Table of Contents

Abstract

Rainfall over a Caribbean island is the residue of a moisture budget whose transport term is barely observed, and the coarse products that drive regional reconstructions carry errors comparable with the signal. This thesis formulates and tests D-LEGOS, a physics-informed method in which space-time is divided into control volumes, each internal face carries one signed transfer booked with opposite signs by its two neighbours, and a weighted budget residual enters the objective. Nine propositions state what such a ledger guarantees (interior transfers cancel exactly) and what it cannot (it is blind to circulation, compatible with wrong transfers, and under exact aggregation inherits a driver’s error in full).

Three hypotheses were tested under decision rules fixed in advance. In a simulation test over 14 to 22°N, 82 to 72°W, with ERA5 as truth, the NCEP/NCAR Reanalysis 1 as background, synthetic observations at the real station sites and 15 test events from 2020 to 2025, the column moisture budget of ERA5 carried information only at coarse scales (its residual was 0.73 of the background’s at 2° and 3 h, and exceeded it at 0.25° and 1 h). Every method that used the ledger lowered the error in block-scale divergence, to 0.81 to 0.83 of the background’s for all but uncoupled blocks (0.91); every method that did not ended above the background. The principal hypothesis, that shared-flux coupling of neural blocks would beat state matching under sparse observations, was not supported: shared-flux blocks, state-matched blocks, a single network and a variational smoother on a grid were indistinguishable on both primary outcomes. No method improved the transport itself by more than 2.5 %. Column-vapour gains were the removal of a large-scale bias and depended more on where stations were than on how many. The weighted ledger kept about half its benefit when the rainfall product under-reported by 50 %.

For monthly rainfall at 0.05° on Jamaica, Hispaniola and Puerto Rico, a lookup table of climatological shares removed 68 % of the uniform-allocation error. Exact aggregation to the coarse driver lowered error by 6 % with an exact driver, and raised it by 15 mm per month relative to a free model when both were fitted to an independent gauge-based driver whose own error was 2.4 times the allocation problem.

The thesis concludes that a weighted budget on coarse control volumes is worth having, that the representation carrying it does not matter, and that conservation should be imposed in proportion to how far the conserved quantity is trusted. All experiments used open gridded data. The application to observed Jamaican events, with station, gauge and sounding data, is specified as a protocol with its calculations marked as awaiting data.

Keywords: physics-informed machine learning; domain decomposition; moisture budget; data assimilation; statistical downscaling; Caribbean rainfall; conservation; observing-system simulation experiment.

Declaration

I declare that this thesis is my own work and that it has not been submitted for a degree at this or any other university. Where the work of others is used it is cited. The experiments reported in Chapters 5 to 7 were designed by me; the code that runs them, the result files and the record of checks are archived with the thesis and listed in Appendix A.

Acknowledgements

[To be written by the candidate.]

Papers arising from this thesis

The thesis is organised so that it can be examined as a monograph or as a set of papers. Four manuscripts accompany it.

None has yet been submitted. Co-authorship and target journals are to be agreed with my supervisors.

What has been computed and what awaits data

How the thesis is organised

Chapter 1 states the problem and the hypotheses. Chapter 2 reviews the literature. Chapter 3 gives the formulation: the physics, the discrete ledger, nine propositions on what the ledger does and does not guarantee, and the inverse problem. Chapter 4 gives the design of the tests and the plan that was frozen before the test events were opened. Chapter 5 reports the simulation test and Chapter 6 the rainfall-allocation experiment. Chapter 7 summarises the component studies. Chapter 8 fixes the protocol for the Jamaican application and marks every calculation that awaits local data. Chapter 9 discusses what the evidence supports, with assumptions, limitations and threats to validity, and Chapter 10 concludes. The appendices cover code and data, numerical checks, the event catalogue and the claim-check record. A separate laboratory manual and an introductory textbook accompany the thesis.

Chapter 1. Introduction

1.1 The problem

On 28 October 2025 Hurricane Melissa made landfall in Jamaica. A global machine-learning model had called a Category 5 landfall five days ahead (Google DeepMind 2026). The rain that fell on the island’s interior was estimated afterwards from a satellite product at 18 to 24 inches over the week (NASA GPM Mission 2025), and that estimate was compared with a forecast, because there was little else to compare it with. The storm was predicted. What it did over the mountains had to be inferred.

That gap is the subject of this thesis. Jamaica sits in a basin where rainfall is set by moisture transport: the trades, the Caribbean low-level jet and the convergence they produce over windward slopes (Taylor et al. 2013; Whyte et al. 2008; Cook and Vizy 2010). The transport is observed almost nowhere. Inside the box from 14 to 22°N and 82 to 72°W, an area of about 940,000 km², the NOAA Integrated Surface Database lists 23 surface stations that reported through 2024 and the IGRA archive lists two radiosonde sites still active, at Kingston and Grand Cayman (Smith, Lott, and Vose 2011; Durre et al. 2018). Three of the surface stations are in Jamaica. Everything else known about the moisture field over that box comes from a global analysis on a grid of about 31 km (Hersbach et al. 2020), and the analysis has its own biases.

A learned model can be given physics to fill the gap. The usual way is a penalty or a constraint that ties the fine field to a coarse driver or to a conservation law. Two things are then taken on trust: that the driver is right, and that satisfying the law makes the field more accurate. Neither holds in general. A constraint that forces agreement with a biased driver copies the bias into every fine cell. A model can balance a water budget to machine precision and still be wrong about the transport that the budget is supposed to describe.

1.2 Research question

The thesis asks one conditional question.

Under which combinations of observation coverage, driver error and interface formulation does physical coupling between local learned models improve the recovered moisture transport over a small-island domain, and under which does it transmit the driver’s error?

Three hypotheses make it testable. They were fixed before any of the experiments that test them were run and are stated in full in Chapter 4.

• H1, exact aggregation under driver bias. With an accurate coarse driver, forcing a fine rainfall field to aggregate exactly to the driver helps or does no harm. As the driver’s bias grows the advantage shrinks and reverses.
• H2, flux coupling under sparse observations. Among local models with matched capacity, a single signed moisture transfer shared by the two blocks on either side of a face recovers withheld transport better than state matching, penalised flux matching or no coupling, and the advantage grows as stations are removed.
• H3, uncertainty. Representing the uncertainty of the background and of the physical closure improves calibrated probabilistic reconstruction over a matched model that treats both as known.

Each hypothesis can fail, and a failure is a result.

1.3 The method in one paragraph

D-LEGOS divides space and time into control volumes (bricks). Inside each brick a small neural network represents the increment to a background analysis. On every face between two bricks there is one number, the water vapour transferred through that face, and the two bricks book it with opposite signs. Internal transfers therefore cancel in the domain total by construction, for any value of the transfers. The reconstruction is then an inverse problem with an objective of the weak-constraint variational kind: observations, a background term, a brick-level water budget with a calibrated tolerance, and an interface term that defines the coupling variant. Chapter 3 gives the formulation. The name comes from the picture of interlocking bricks. The science does not depend on the picture.

1.4 What the thesis contributes

1. A formulation with its limits stated as propositions. The ledger identity, and seven further statements about what it cannot deliver: accuracy, local uncertainty, the rotational part of the transport, immunity to driver bias, closure across scales (Chapter 3).
2. A measured closure floor. For the reference analysis itself, the column water-vapour budget closes no better than a ten-times-coarser background at the scale of one grid cell and one hour, and becomes informative only for control volumes of about two degrees and three to six hours (Chapter 5). Any physics-informed constraint on this budget inherits that floor.
3. An executed simulation test of flux coupling on fifteen held-out events in five regimes, including Hurricanes Beryl (2024) and Melissa (2025), with an independent reanalysis as the biased background and the real station geometry (Chapter 5).
4. A controlled test of exact aggregation under driver error on three Caribbean islands, with a learned model that predicts departures from the seasonal pattern, a signed bias sweep and an independent gauge-based driver (Chapter 6).
5. A set of executed component studies that fix the baselines any Caribbean rainfall or temperature model must beat, several of them negative results (Chapter 7).
6. A protocol for the Jamaican application that states which calculations need Meteorological Service of Jamaica data and what would count as success (Chapter 8).

1.5 Evidence classes

Results in this thesis come from four kinds of evidence and each result is labelled with its kind.

• Derived: propositions with proofs, each rechecked numerically.
• Simulation test: a nature run stands in for truth, synthetic observations are drawn from it at real station sites, and recovery is measured exactly. The nature run (ERA5) and the background (NCEP/NCAR Reanalysis 1) come from different models and assimilation systems.
• Observed product: experiments scored against a satellite–gauge analysis (CHIRPS) or a reanalysis. Skill against a product is skill against that product.
• In-situ: experiments scored against station measurements. Only two component studies have any, and neither has Jamaican gauges that are independent of the reference product.

No result in this thesis is a claim of operational forecast skill.

1.6 Publications and structure

The thesis is organised around four papers, following the consolidation of nine earlier drafts.

Chapter 2 reviews the literature. Chapter 3 gives the formulation and Chapter 4 the research design. Chapters 5 to 7 report results. Chapter 8 sets out the Jamaican application and marks every calculation that awaits local data. Chapter 9 discusses limits, assumptions and threats to validity, and Chapter 10 concludes.

Chapter 2. Literature review

2.1 Scope and argument

Every learned method for local climate information over a small island conditions on a coarse field and, increasingly, on a physical law. This review argues that the literature has established the two halves of a problem and has not put them together. The regional record shows that the coarse field over the Caribbean is biased for structural reasons. The machine-learning record shows that constraints which enforce agreement with a field, or closure of a budget, are satisfied by construction or by optimisation. No study treats the error of the constrained quantity as the experimental variable, and no study asks what a conservation identity between local models does for the accuracy of a field that almost nobody observes. Those two omissions are what Chapters 5 and 6 test.

The review covers seven bodies of work in the order the argument needs them: Caribbean rainfall and the transport that sets it (2.2); the coarse drivers and their errors, including the closure of analysed moisture budgets (2.3); data-driven weather models and what they inherit (2.4); the ways physics enters learning (2.5); domain decomposition (2.6); downscaling under driver error (2.7); and data assimilation, to which the reconstruction problem belongs (2.8). Section 2.9 reviews verification practice and Section 2.10 maps the gaps onto the hypotheses.

Sources come from the thesis reference register, which records for each entry whether the full text or only the abstract was read. Entries read at abstract level are cited here only for what their abstracts state. This chapter is the synthesis used by the thesis; a longer treatment with the full register is held with the project records.

2.2 Caribbean rainfall and the transport that sets it

2.2.1 The annual cycle and its mechanism

Jamaican rainfall is bimodal. An early wet season in May and June and a main season centred on October are separated by a drier July, the midsummer drought, first described for Mexico and Central America by Magaña, Amador, and Medina (1999) and placed in the Caribbean climatology by Gamble and Curtis (2008). The 1951–1980 parish climatology gives the amplitude: the island mean rises from 83 mm in March to 226 mm in May, falls to 135 mm in July and peaks at 271 mm in October (Climate Studies Group Mona 2012). The July mean hides a parish range from 50 mm to 231 mm, so a single island figure describes no place on the island.

The agent of the midsummer minimum is the Caribbean low-level jet, an intensification of the easterly trades between 70 and 80°W with its axis near 15°N, confined below about 600 hPa, with speeds approaching 16 m s⁻¹ at 925 hPa and a peak in July (Whyte et al. 2008; Muñoz et al. 2008; C. Wang 2007). Cook and Vizy (2010) analyse its dynamics and its link to precipitation. The jet matters to this thesis for a specific reason: the summer drying acts through transport and shear, and the same mechanism is the leading explanation of the drying that climate models project for the basin (Taylor et al. 2013). A model of Jamaican rainfall that omits the lateral moisture flux has removed the variable through which both act.

2.2.2 Ocean controls and non-stationarity

Interannual variability follows sea surface temperature in two basins. Giannini, Kushnir, and Cane (2000) established the El Niño–Southern Oscillation as the dominant interannual mode, acting partly through the tropical Atlantic, and Enfield and Alfaro (1999) and Taylor, Enfield, and Chen (2002) separated the roles of the two oceans. The sign of the ENSO relationship changes within the year: positive for the early season at a lag of several months (Chen and Taylor 2002), negative for the late season (Peterson et al. 2002).

These relationships are moving. Clarke et al. (2026) report that tropical North Atlantic sea surface temperature, long the reliable predictor of early-season rainfall, now crosses the convective threshold earlier in the year without more early rain, and that dynamic variables explain a larger share of the variance than absolute temperature. I read that result as a warning about any constraint tuned to a stationary relationship between a driver and local rainfall.

2.2.3 The observed record and the observing network

Two workshop analyses anchor the record of extremes. Peterson et al. (2002) used 30 stations for 1958–1999 and Stephenson et al. (2014) 51 stations for 1961–2010. Both find warming in the temperature indices. Precipitation trends are weak, and the sign of the trend in consecutive dry days reverses between the two studies. Jamaica contributed two stations to the second analysis, and its series are short for a documented reason: a 1992 fire destroyed a large part of the national archive (Peterson et al. 2002).

The premise that observations are sparse needs numbers. Table 2.1 counts the network inside the analysis domain used in Chapter 5 from two public archives.

Table 2.1. Observing network in the domain 14–22°N, 82–72°W (about 940,000 km²; 1,280 cells of the 0.25° analysis grid, 117 of them more than half land). Counts are from the NOAA ISD station history and the IGRA v2 station list as retrieved on 11 November 2026. Rain-gauge counts held by the Meteorological Service of Jamaica, and the number of those gauges that enter CHIRPS, are not in either archive and are an open item (Chapter 8).

At two-degree blocks the domain holds 20 blocks and the 23 stations fall in 9 of them. Eleven blocks, most of the open sea across which the jet carries its moisture, hold no station at all.

The background analysis sees little of the island. On the 0.25° ERA5 grid 16 cells over Jamaica are more than half land, and the highest model surface there is 373 m, where the standard atmosphere pressure is about 969 hPa. Blue Mountain Peak stands at 2,256 m. The terrain that forces the island’s rainfall contrast is a low hill in the analysis, and no standard pressure level above 975 hPa intersects it.

2.3 The coarse driver and its errors

2.3.1 Global and regional models

A global climate model represents Jamaica by one or two grid boxes, and the 50 km PRECIS grid by twelve, every one a mixture of land and sea (Climate Studies Group Mona 2012). Ryu and Hayhoe (2014) trace the Caribbean precipitation biases of CMIP3 and CMIP5 models to errors in sea surface temperature and circulation. Regional models inherit and add. Driven by reanalysis, so that the driver is as good as it can be, PRECIS at 50 km underestimates precipitation over most of the domain and most strongly over the large islands, with surface moisture fluxes 10 to 20 % below the reanalysis, and the dry bias appears whatever the domain size (Centella-Artola et al. 2015). Driven by a global model it simulates 1 to 2 mm day⁻¹ too little rain over Cuba, Hispaniola, Jamaica and Puerto Rico in every season (Campbell et al. 2011). With RegCM at the same grid, changing the convective closure changes the sign of the island rainfall bias (Martı́nez-Castro et al. 2016). The lesson I take from the three studies is that the bias of a coarse driver over these islands is set by choices inside the model that a downstream method cannot observe.

Finer grids have not settled the matter. A 10 km simulation over the Lesser Antilles separates land from sea where its driver sees only sea, and its authors still call the precipitation projections less reliable because the seasonal cycle is not reproduced (Cantet et al. 2014). An 8 km WRF simulation reduces the wind bias at Sangster and not the temperature bias, on an evaluation that only four stations across three islands survived (Williams et al. 2026).

2.3.2 Reference products

The products used as truth disagree among themselves. Centella-Artola et al. (2020) compared sixteen gridded precipitation datasets with Caribbean gauges and found MSWEP (Beck et al. 2019) the best performer. CHIRPS (Funk et al. 2015) blends infrared cold-cloud estimates with stations at 0.05°, with weights that assume a correlation of 0.5 between the satellite estimate and true rainfall where no station is near. Two consequences follow for this thesis. A station that enters CHIRPS cannot verify a model trained on CHIRPS. Where there is no station, the product is a satellite proxy at a nominal resolution finer than its information.

Gauge-only analyses have the opposite character. The GPCC Full Data Monthly product (Schneider et al. 2018) interpolates quality-controlled gauge totals to grids as fine as 0.25° and uses no satellite input, so over an island it is as good as the number of gauges that reported that month, and it is silent on structure between them. Because its errors come from a different source than those of a satellite blend, it serves in Chapter 6 as an independent coarse driver: a driver that is wrong in a realistic way and not by construction.

Reanalyses differ by generation as much as by centre. The NCEP/NCAR Reanalysis 1 (Kalnay et al. 1996) runs a model of about 210 km grid spacing with a 1990s assimilation system; ERA5 (Hersbach et al. 2020) runs at 31 km with hourly output and four-dimensional variational assimilation; MERRA-2 (Gelaro et al. 2017) sits between them. Over the Caribbean the older product resolves the trades and the low-level jet only coarsely, and its grid has no Jamaica. That contrast is what the simulation test of Chapter 4 exploits: the older reanalysis plays the part of an imperfect driver and the newer one the part of the truth, so the size and structure of the driver error are those of a real pair of products.

2.3.3 Closure of analysed moisture budgets

The column water-vapour budget links transport to rainfall, and analysed products do not close it. Trenberth and Guillemot (1995) evaluated the global moisture budget from analyses and documented residuals that arise from the analysis increments and from sampling. Seager and Henderson (2013) computed moisture budgets from the ERA-Interim archive and showed that the result depends on the vertical and temporal discretisation used, to the extent that the divergence term must be computed with care on the model’s own levels before the residual is small enough to interpret. ERA5 (Hersbach et al. 2020) distributes the vertically integrated vapour transport computed on model levels, which removes the vertical part of that problem and leaves the horizontal, temporal and assimilation parts.

The implication is practical. A physics-informed penalty on this budget asks a learned field to close an equation that its own training target does not close. The size of the target’s residual, at the control volume the penalty uses, is the floor below which the penalty carries no information. I found no study in the machine-learning literature reviewed here that reports that floor before imposing the penalty. Chapter 5 reports it.

2.4 Data-driven weather models and what they inherit

Between 2022 and 2026 a family of global models learned the atmospheric time step from reanalysis. Deterministic models at 0.25° (Bi et al. 2023; Lam et al. 2023) matched or beat operational forecasts on most upper-air scores. Probabilistic training followed, by diffusion (Price et al. 2025) and by proper scores optimised directly (Lang et al. 2024). Three findings from that literature bear on this thesis.

Precipitation is the weakest variable, and its benchmark truth is weak. WeatherBench 2 uses ERA5 precipitation as ground truth while stating that it is not directly assimilated and can differ widely from gauges, and it replaces root-mean-square error by a categorical score for precipitation because averaged scores favour unrealistically smooth forecasts (Rasp et al. 2024).

Models trained on an analysis inherit its biases, and the developers say so. The WeatherNext 3 preprint states that analysis-trained models cannot use observations directly and that biases in the analysis are inherited by the forecast (Rasp et al. 2026). The cyclone system built on the same line attributes the low-intensity bias of earlier models to the 0.25° analysis and removes it by training on best-track data (Alet et al. 2026). The remedy in both cases was a better target, not a finer grid.

A budget constraint can remove part of the inheritance. Yuval et al. (2025) train a hybrid general circulation model (Kochkov et al. 2024) on satellite precipitation under an enforced column water budget and report better precipitation scores than an operational ensemble at 2.8°, while noting that the model still underestimates the most extreme rates. At that resolution the model cannot see Jamaica. The result I carry forward is narrower: a column budget used as a constraint has been shown to help when the target is an observational product and the scale is hundreds of kilometres.

Mozaffari et al. (2026) argue that this literature concentrates skill where observations are dense, because a mean-square loss descends fastest there. The Caribbean network of Table 2.1 is the kind of place their argument is about.

2.5 How physics enters learning

2.5.1 Soft, hard and structural

Physics-informed machine learning covers any method in which a differential equation, a conservation law or a known relation between scales shapes the training or the architecture (Karniadakis et al. 2021). Three routes matter here. A soft constraint is a weighted penalty, as in physics-informed neural networks (Raissi, Perdikaris, and Karniadakis 2019); it can be violated at test time and carries no guarantee (Kashinath et al. 2021). A hard constraint is an architectural layer that satisfies a relation exactly: Beucler et al. (2021) enforce conservation in a convection emulator, and Harder et al. (2023) renormalise a downscaled field so that its coarse aggregate equals the input. A structural constraint builds the law into the function class. Richter-Powell, Lipman, and Chen (2022) parameterise divergence-free fields so that a continuity equation holds by construction, and Hansen et al. (2023) enforce an integral conservation law on a probabilistic model as a constraint on its predictive distribution.

The shared-face transfer of D-LEGOS belongs to the third family and is closest in spirit to the last two. It differs in what is conserved and where: the conserved quantity is the vapour in a control volume, the constraint is an algebraic identity between neighbouring local models and not a property of one global network, and the source term on the right-hand side is an uncertain observation product. Neither of the two neural conservation papers has a biased forcing or an interface between models.

2.5.2 Failure modes

Soft penalties fail for reasons that are now understood. Krishnapriyan et al. (2021) show a standard physics-informed network reaching relative errors near 100 % on a one-dimensional convection problem that a classical solver handles trivially, and trace the failure to the conditioning of the loss, not to capacity. S. Wang, Teng, and Perdikaris (2021) identify imbalance between the gradients of the residual and boundary terms, and S. Wang, Sankaran, and Perdikaris (2024) show that minimising residuals at all times at once violates causality. I met a version of the first problem in the development runs of Chapter 4: with a poorly conditioned prior the block networks fitted station noise exactly while the objective kept falling. A failed optimisation is not evidence against a constraint, and a converged one is not evidence for it.

Whether a soft physics penalty improves generalisation is contested inside this literature. Beucler et al. (2021) and Harder et al. (2023) report gains. Under a controlled comparison on ocean problems, an added physics loss degraded accuracy on unseen data for every activation tried (Wolff et al. 2021). The disagreement is the reason this thesis reports skill and closure as separate outcomes.

2.6 Domain decomposition

Decomposition attacks the growth of the optimisation problem with domain size and the spectral bias of a single network. Two families differ in how subdomains communicate. In conservative and extended physics-informed networks the domain is cut into non-overlapping subdomains with one network each, and interface conditions are penalties: continuity of the solution and, in the conservative form, of the normal flux (Jagtap, Kharazmi, and Karniadakis 2020; Jagtap and Karniadakis 2020). Because the conditions are penalties, the assembled solution is discontinuous to the extent the penalty is not driven to zero. Finite-basis networks remove the interface loss by multiplying each local network by a smooth window and summing over overlapping subdomains (Moseley, Markham, and Nissen-Meyer 2023), and a multilevel extension adds coarse levels that carry the low frequencies (Dolean et al. 2024).

Two results from this literature shaped the design. The first is the trade-off of Hu et al. (2022): a decomposition helps only if each local target becomes simpler by more than the local data become fewer. Applied to Table 2.1, two-degree blocks leave eleven of twenty blocks with no observation, so the local data in most blocks are none and any gain must come through the coupling or through a shared coarse component. The second is the multilevel result itself. One-level decompositions scale poorly because nothing carries information across the domain, and the fix is a coarse level. The baseplate of Chapter 3 is the smallest such level, twelve parameters, and the development runs showed it to be necessary.

None of the decomposition papers has a driver, a biased forcing or sparse observations, and none asks whether the assembled field satisfies a budget. The flux penalty of the conservative form is the nearest precedent for the variants compared in Chapter 5, where it appears as variant C2.

2.7 Downscaling under driver error

Neural downscaling began as image super-resolution (Vandal et al. 2017), moved to adversarial training to restore small-scale variance (Stengel et al. 2020; Leinonen, Nerini, and Berne 2021) and then to diffusion (Addison et al. 2022). Two regularities recur. On root-mean-square error a model trained on that error wins or ties, and on spectra and proper scores the generative model wins. And nearly every study builds its coarse input by coarsening its own fine reference, so the driver is exact by construction.

Four studies show what happens when it is not. Addison et al. (2022) could not train on the global model’s output because its weather differed from the fine model’s at the same time stamp, and saw the bias of their emulator change sign over uplands when they sampled from it. Meer, Roda Husman, and Lhermitte (2023) trained one emulator on coarsened fine-model fields and one on the real coarse model: the first reproduced the fine model when given coarsened input and consistently underestimated it when given the real driver. Price and Rasp (2022) found that pure super-resolution trained on coarsened truth scored no better than interpolation on real forecasts. Harder et al. (2023) give the cleanest case. Their hard layer conserves the coarse total to rounding error and improves skill across architectures when the coarse field is pooled from the fine one; on a pair of separately simulated fields, constraining lowered predictive skill.

The four observations describe one mechanism, which Proposition 7 of Chapter 3 states as a bound. None of the four isolates it, because each compares two drivers that differ in unknown ways. The nearest recent method adds soft conservation penalties to a flow-matching downscaler and tests out of distribution by changing region (Debeire et al. 2026). A change of region is not a controlled perturbation of the driver.

The executed component study behind Paper B had already found the practical form of the problem on Jamaican data: exact conservation improved a matched network and both lost to a twelve-month lookup of climatological shares. Chapter 6 extends it.

2.8 Data assimilation and its hybrids

Reconstructing a state from sparse observations, a background and a model is data assimilation, and a thesis that proposes to do it with neural fields has to say what is new. Carrassi et al. (2018) review the field. The weak-constraint four-dimensional variational method (Trémolet 2006) minimises a sum of background, observation and model-error terms over a window, which is the objective of Chapter 3 with the model error evaluated as a ledger residual. Simulation experiments in which synthetic observations are drawn from a nature run are the standard way to measure recovery when truth is otherwise unknown, with the known hazard that a nature run from the same system as the background flatters the result (Hoffman and Atlas 2016).

Machine learning has entered assimilation from two sides. Brajard et al. (2020) alternate assimilation and learning to emulate a dynamical model from sparse, noisy observations. Rozet and Louppe (2023) sample state trajectories given observations from a learned score. Both learn the dynamics or the prior. D-LEGOS learns neither. It fits fields to one window, and what it changes relative to the variational method is the parameterisation of the state.

A neural-field parameterisation offers four concrete properties over a grid of values. Observation operators are mesh-free: a station is evaluated at its own coordinates. The fields are differentiable in space and time, so a flux on a face is evaluated, not interpolated. Parameters could be shared across events. And the cost profile differs. Chapter 5 tests the first two against a grid-based variational smoother given the same observations, the same covariance and the same ledger, and reports the fourth. The third is left for later work. If the grid smoother does as well, the honest conclusion is that the parameterisation is a convenience and not a source of skill.

2.9 Verification practice

Four hazards recur in the literature above and the design answers each. Scores that reward blur: mean-square error favours smooth fields, so proper scores for distributions (Gneiting and Raftery 2007; Ferro 2014) and for joint structure (Scheuerer and Hamill 2015) are reported beside it, and probabilistic output is scored against reference distributions and never against point forecasts alone, a lesson one of the component studies learned the hard way (Chapter 7). Dependence: random splits of structured data underestimate error (Roberts et al. 2017), so splits are by whole event or by year and intervals come from a block bootstrap over those units (Künsch 1989). Leakage through the truth product: a model trained on a product scores best against that product (Rasp et al. 2026). Diagnostics that repeat the objective: a budget residual computed with the training operators measures the fit, so every physical diagnostic in Chapter 5 is recomputed from the assembled field with a different discretisation.

2.10 Synthesis

Table 2.2. What each strand establishes, what it leaves open, and where this thesis addresses it.

The contribution is the test. The identities are textbook, the constraint layers exist, local networks exist. What had not been done is to measure the closure floor of the reference, to hold observation coverage and driver error as controlled factors, and to give the conventional method the same information.

Chapter 3. Formulation: the physics, the ledger and the inverse problem

This chapter states D-LEGOS in full. Section 3.1 fixes notation. Sections 3.2 and 3.3 derive the physical statement the method rests on, the column water-vapour budget, from the hydrostatic primitive equations. Sections 3.4 and 3.5 turn it into a discrete ledger on space-time control volumes and collect, as propositions with proofs, what that ledger guarantees and what it cannot. Sections 3.6 to 3.9 pose the reconstruction as an inverse problem: the parameterisation, the observation operators, the objective, the four coupling variants and the baselines. Section 3.10 defines the uncertainty construction, Section 3.11 the rainfall-allocation sub-problem, and Section 3.12 the pressure-level class that the column form reduces. Every symbol in this chapter has a counterpart in the code of Appendix A, and Table 3.4 gives the mapping.

Three levels of status apply and I keep them apart. Derivations and propositions are complete and each is checked numerically (Appendix B). The column form of Sections 3.6 to 3.10 is implemented and has been run (Chapter 5). The pressure-level class of Section 3.12 is specified and has not been run.

3.1 Notation and units

Table 3.1. Symbols and units.

Geopotential and transfer both carry the letter Φ in the literature. I keep both and distinguish the transfer by its face subscript. Hourly accumulations of E and P are quoted in kg m⁻², numerically equal to millimetres.

3.2 The resolved atmosphere

The reconstruction class is hydrostatic and uses pressure as the vertical coordinate. With D/Dt = ∂_t + v ⋅ ∇_p + ω ∂_p, Coriolis parameter f, and κ = R_d/c_p, the resolved balances are

$$\frac{D\mathbf{v}}{Dt} + f\,\mathbf{k} \times \mathbf{v} = - \nabla_{p}\Phi + \mathbf{F}_{h},\quad\quad\frac{\partial\Phi}{\partial p} = - \frac{R_{d}T_{v}}{p},\quad\quad\nabla_{p} \cdot \mathbf{v} + \frac{\partial\omega}{\partial p} = 0,\quad\quad\text{(3.1)}$$

$$\frac{DT}{Dt} - \frac{\kappa T\omega}{p} = \frac{\dot{Q}}{c_{p}},\quad\quad\frac{Dq}{Dt} = S_{q}.\quad\quad\text{(3.2)}$$

F_h, Q̇ and S_q are the unresolved momentum tendency, the diabatic heating rate and the net humidity source. These are the equations documented for the ECMWF Integrated Forecasting System (ECMWF 2023), reduced to pressure coordinates in the standard way (Holton and Hakim 2013). Hydrostatic balance removes vertical acceleration, so explicit deep convection is outside the class. A fine output grid does not change that.

Only the last equation of (3.2) and the continuity equation are needed for the column form. The others return in Section 3.12.

3.3 The column water-vapour budget

Multiply the humidity equation by 1/g, use continuity to write advection in flux form, Dq/Dt = ∂_tq + ∇_p ⋅ (qv) + ∂_p(qω), and integrate from the model top p_t to the surface pressure p_s(λ, φ, t):

$$\frac{1}{g}\int_{p_{t}}^{p_{s}}\left\lbrack \partial_{t}q + \nabla_{p} \cdot \left( q\mathbf{v} \right) + \partial_{p}(q\omega) \right\rbrack dp = \frac{1}{g}\int_{p_{t}}^{p_{s}} S_{q}\, dp.\quad\quad\text{(3.3)}$$

The lower limit moves, so the time derivative and the divergence do not commute with the integral. Leibniz’s rule gives

$$\frac{1}{g}\int_{p_{t}}^{p_{s}}\partial_{t}q\, dp = \partial_{t}W - \frac{q_{s}}{g}\,\partial_{t}p_{s},\quad\quad\frac{1}{g}\int_{p_{t}}^{p_{s}}\nabla_{p} \cdot \left( q\mathbf{v} \right)\, dp = \nabla \cdot \mathbf{Q} - \frac{q_{s}}{g}\,\mathbf{v}_{s} \cdot \nabla p_{s},\quad\quad\text{(3.4)}$$

with

$$W = \frac{1}{g}\int_{p_{t}}^{p_{s}}q\, dp,\quad\quad\mathbf{Q} = \frac{1}{g}\int_{p_{t}}^{p_{s}}q\,\mathbf{v}\, dp.\quad\quad\text{(3.5)}$$

The vertical term integrates to (q_sω_s − q_tω_t)/g. At the surface the kinematic condition is ω_s = ∂_tp_s + v_s ⋅ ∇p_s, so the three surface terms cancel exactly. With negligible vapour flux through the top, and with the column integral of S_q equal to surface evaporation minus the net conversion of vapour to condensate, which I write as E − P under the assumption that condensate is neither stored nor exported, the result is

$$\frac{\partial W}{\partial t} + \nabla \cdot \mathbf{Q} = E - P.\quad\quad\text{(3.6)}$$

On the sphere ∇ ⋅ Q = (acos φ)⁻¹[∂_λQ_λ + ∂_φ(Q_φcos φ)].

Equation (3.6) is exact only under four assumptions, and each is a named source of residual when the terms come from an analysed product:

1. condensate storage and transport are neglected, so rain that falls from cloud water formed earlier or elsewhere is booked in the wrong hour or cell;
2. vapour flux through the model top is neglected;
3. the integral runs from the true surface pressure, which pressure-level data resolve poorly over terrain;
4. the fields obey the model equations between analysis times, which an analysis does not, because each assimilation cycle adds an increment that is a source of water in the budget (Seager and Henderson 2013; Trenberth and Guillemot 1995).

For that reason no reconstruction in this thesis is credited with closing the budget until the residual of the reference product itself has been measured with the same operators. Chapter 5 does this for ERA5 and finds the size of control volume below which the budget carries no usable information.

3.4 Space-time control volumes and the discrete ledger

3.4.1 Integral form

Let Ω_i be a spatial control volume with boundary faces f and I_n = [t_n, t_n + 1] a time window. The brick is K_iⁿ = Ω_i × I_n. Integrating (3.6) over the brick and applying the divergence theorem,

$$\underset{M_{i}^{n + 1} - M_{i}^{n}}{\underbrace{\int_{\Omega_{i}}^{} W\left( t_{n + 1} \right)\, dA - \int_{\Omega_{i}}^{} W\left( t_{n} \right)\, dA}} + \sum_{f \subset \partial\Omega_{i}}^{}s_{i,f}\underset{\Phi_{f}^{n}\,\Delta t}{\underbrace{\int_{I_{n}}^{}\int_{f}^{}\mathbf{Q} \cdot \mathbf{n}_{f}\, dl\, dt}} = \underset{S_{i}^{n}}{\underbrace{\int_{I_{n}}^{}\int_{\Omega_{i}}^{}(E - P)\, dA\, dt}}.\quad\quad\text{(3.7)}$$

Each internal face is oriented once, with unit normal n_f pointing east or north, and s_i, f = +1 for the block on the western or southern side and −1 for the other. One number Φ_fⁿ per face, booked with opposite signs by its two blocks, is the ledger.

3.4.2 Geometry on the sphere

Cells are centred on the nodes of a regular latitude–longitude grid with spacings Δλ and Δφ. The cell with centre latitude φ_j has exact area

$$A_{j} = a^{2}\,\Delta\lambda\,\left\lbrack \sin\left( \varphi_{j} + \frac{1}{2}\Delta\varphi \right) - \sin\left( \varphi_{j} - \frac{1}{2}\Delta\varphi \right) \right\rbrack.\quad\quad\text{(3.8)}$$

Faces crossed by the eastward transport are meridian segments of length ℓ^u = a Δφ. Faces crossed by the northward transport are parallel segments of length ℓ_j + 1/2^v = acos φ_j + 1/2 Δλ. The finite-volume divergence of a cell is

$$\left( \nabla \cdot \mathbf{Q} \right)_{j,i} = \frac{1}{A_{j}}\lbrack\ell^{u}\left( Q_{\lambda;\, j,i + 1/2} - Q_{\lambda;\, j,i - 1/2} \right) + \ell_{j + 1/2}^{v}Q_{\varphi;\, j + 1/2,i} - \ell_{j - 1/2}^{v}Q_{\varphi;\, j - 1/2,i}\rbrack.\quad\quad\text{(3.9)}$$

Because the face lengths and areas are exact, a uniform eastward transport has zero discrete divergence to rounding error and a uniform northward transport returns −Q_φtan φ/a, the convergence of the meridians. Both are unit tests (Appendix B).

3.4.3 The block ledger used in this thesis

A block is a rectangle of b × b cells and a ledger window is w hours. With hourly time levels t_h, the trapezoidal rule in time and midpoint quadrature along each face, the residual of block i in window n is

$$R_{i}^{n} = \frac{1}{A_{i}\, w}\left\lbrack M_{i}\left( t_{n + 1} \right) - M_{i}\left( t_{n} \right) + \sum_{h}^{}\frac{\Delta t}{2}\left( F_{i}\left( t_{h} \right) + F_{i}\left( t_{h + 1} \right) \right) - S_{i}^{n} \right\rbrack,\quad\quad\text{(3.10)}$$

with the net outflow through the block edge

$$F_{i}(t) = \sum_{f \subset \partial\Omega_{i}}^{}s_{i,f}\sum_{m \in f}^{}\ell_{m}\, Q_{n}\left( \mathbf{x}_{m},t \right).$$

$M_{i} = \sum_{c \in \Omega_{i}}^{}A_{c}W_{c}$ is the stored vapour, Q_n is the transport component normal to the face, m runs over the cell-face midpoints along the block edge and Δt = 3600 s. R_iⁿ is in kg m⁻² per hour, so residuals from blocks and windows of different size can be compared.

3.5 What the ledger guarantees and what it cannot

Collect the block masses in a vector M ∈ ℝ^N and the internal transfers in Φ ∈ ℝ^F. Let B ∈ ℝ^N × F have +1 at the receiving block and −1 at the sending block of each internal face. With boundary transfers b and sources s,

$$\frac{d\mathbf{M}}{dt} = \mathsf{B}\mathbf{\Phi} + \mathbf{b} + \mathbf{s},\quad\quad\mathbf{1}^{\top}\mathsf{B} = \mathbf{0}^{\top}.\quad\quad\text{(3.11)}$$

Proposition 1 (samplewise cancellation). For any transfer vector Φ whatsoever, 1^⊤ dM/dt = 1^⊤(b + s). Internal transfers contribute nothing to the domain total, for every ensemble member and not only in expectation.

Proof. Each column of B holds one +1 and one −1. ▫

The identity holds for the discrete update only if one time-integrated transfer is used on both sides of a face. Two separate quadratures of the two sides break it. It says nothing about whether Φ is right.

Proposition 2 (dissipation is separate from conservation). For the reduced exchange law Φ = −diag(G) B^⊤T with non-negative conductances and storage M = CT, the capacity-weighted variance $V = \frac{1}{2}\sum_{i}^{}C_{i}\left( T_{i} - \mu \right)^{2}$ about the conserved mean μ obeys $dV/dt = - \sum_{e}^{}G_{e}\left( \Delta T_{e} \right)^{2} \leq 0$.

Proof. μ is invariant because 1^⊤B = 0^⊤. Differentiate V, substitute C dT/dt = −B diag(G) B^⊤T and note that the constant vector is annihilated by B^⊤. ▫

The total is constant while differences decay. A sign error can preserve the first and violate the second, which is why both are tested.

Proposition 3 (local uncertainty depends on cross-face covariance). For an ensemble with fixed boundary and source terms, Cov(dM/dt) = B Σ_Φ B^⊤. For a block with two opposite faces, Var(Ṁ) = Var(Φ₁) + Var(Φ₂) − 2 Cov(Φ₁, Φ₂).

Sampling the faces independently keeps the ledger exact and still mis-states the spread of every block tendency by the omitted covariance.

Proposition 4 (cycle ambiguity). If observations constrain only the storage trajectory and the external terms, any transfer perturbation c with Bc = 0 leaves the trajectory unchanged. On a connected graph the space of such c has dimension F − N + 1. For a lattice of n_y × n_x blocks this is (n_y − 1)(n_x − 1).

Proof. The first statement is (3.11). A connected graph has rank B = N − 1, so dim ker B = F − N + 1. The lattice has F = n_y(n_x − 1) + n_x(n_y − 1) internal faces. ▫

The 4 × 5 lattice of two-degree blocks used in Chapter 5 has 31 internal faces and a twelve-dimensional space of circulating transfers that no storage observation can see. In continuous terms, (3.6) constrains ∇ ⋅ Q and is silent about ∇ × Q. Only direct transport information (winds, humidity profiles) or the prior constrains the rotational part. Chapter 5 reports the curl error of every method for this reason.

Proposition 5 (false precision from a biased background). With truth m^⋆, observation y = m^⋆ + ϵ, ϵ ∼ 𝒩(0, σ²), and a Gaussian prior with mean m^⋆ + δ and variance τ², the posterior mean is m̂ = wy + (1 − w)(m^⋆ + δ) with w = τ²/(τ² + σ²). Its reported variance is τ²σ²/(τ² + σ²), its bias is (1 − w)δ and its mean-square error is w²σ² + (1 − w)²δ².

As τ → 0 the reported variance goes to zero and the error to δ. A narrow interval under a confident, biased background is no evidence of accuracy.

Proposition 6 (constraint compatibility across scales). Let 𝒜 aggregate a fine state to a coarse one, 𝒞_h and 𝒞_H be linear constraint operators at the two scales and 𝒜_C a bounded map between their outputs. With the defect e_C(x) = 𝒞_H𝒜x − 𝒜_C𝒞_hx, a fine state with 𝒞_hx = 0 has coarse residual e_C(x), and in general  ∥ 𝒞_H𝒜x ∥  ≤  ∥ e_C(x) ∥ + ∥ 𝒜_C∥  ∥ 𝒞_hx∥.

A coarse residual can therefore arise from incompatibility of the two discretisations alone.

Proposition 7 (error inherited from a biased driver under exact aggregation). Let $C_{c}\lbrack p\rbrack = \sum_{j \in c}^{}a_{j}p_{j}/A_{c}$ be the area-weighted mean over coarse cell c, p^⋆ the true fine field, and let the coarse driver be d_c = C_c[p^⋆] + β_c. Any estimate p̂ that satisfies C_c[p̂] = d_c for every c has

$$\frac{\sum_{j}^{}a_{j}\left( {\widehat{p}}_{j} - p_{j}^{\star} \right)^{2}}{\sum_{j}^{}a_{j}}\ \geq \ \frac{\sum_{c}^{}A_{c}\beta_{c}^{2}}{\sum_{c}^{}A_{c}}.\quad\quad\text{(3.12)}$$

For a multiplicative bias d_c = (1 + α)C_c[p^⋆] the bound is α² times the area-weighted mean of C_c[p^⋆]².

Proof. Linearity gives C_c[p̂ − p^⋆] = β_c. By Jensen’s inequality within each coarse cell, $\sum_{j \in c}^{}a_{j}\left( {\widehat{p}}_{j} - p_{j}^{\star} \right)^{2} \geq A_{c}\beta_{c}^{2}$. Sum over c. ▫

Equality holds when the error is uniform within each coarse cell. In general the mean-square error splits as the floor plus the within-cell variance of the error, a decomposition that Chapter 6 checks to rounding error.

Proposition 8 (assembled closure). Let e_i be the ledger residual of block i computed with its own one-sided transfers and m_f = Φ_f⁽ⁱ⁾ − Φ_f^(j) the mismatch of the two one-sided estimates on internal face f. The residual of the domain budget is $R_{\Omega} = \sum_{i}^{}e_{i} - \sum_{f}^{}m_{f}$, so $|R_{\Omega}| \leq \sum_{i}^{}|e_{i}| + \sum_{f}^{}|m_{f}|$.

Local closure and interface agreement are separate quantities and both must be measured. With one shared transfer per face, m_f = 0 identically.

Proposition 9 (blending is not conserving). For a partition of unity {w_b} and block transport fields Q_b, $\nabla \cdot \left( \sum_{b}^{}w_{b}\mathbf{Q}_{b} \right) = \sum_{b}^{}w_{b}\,\nabla \cdot \mathbf{Q}_{b} + \sum_{b}^{}\nabla w_{b} \cdot \mathbf{Q}_{b}$. The second sum vanishes only where the block fields agree, because $\sum_{b}^{}\nabla w_{b} = 0$.

A smooth blended field can look seamless and still carry a spurious source in every overlap. This is the reason the thesis uses hard block cores and measures transfers on faces, where finite-basis methods blend (Moseley, Markham, and Nissen-Meyer 2023).

3.6 The reconstruction as an inverse problem

3.6.1 Task

The task is smoothing: estimate the state over a window from observations before, during and after each time, p(x_0 : N ∣ y_0 : N). Its scores cannot be compared with those of a forecast, which may use only y_0 : t.

3.6.2 State and parameterisation

The state of the column form is x = (W, Q_λ, Q_φ) as functions of position and time. Each component k is written as a background plus two learned increments,

x^(k)(λ, φ, t) = x_b^(k)(λ, φ, t) + s_c^(k) a^(k) ⋅ ϕ(λ̃, φ̃, t̃) + s_s^(k) f_θi^(k)(ξ, η, τ),    (λ, φ) ∈ Ω_i.    (3.13)

Background. x_b is the large-scale driver interpolated to the point: bilinear in space, linear in time.

Baseplate. ϕ = (1, λ̃, φ̃, t̃) in domain coordinates scaled to [−1, 1], with four coefficients per component, twelve in all, shared by every block. It carries the domain-wide part of the background error. Block-local fields cannot correct that part in a block that holds no observation, which is what the development runs showed before the plan was frozen (Chapter 4).

Block fields. f_θi : ℝ⁵ → ℝ³ is a multilayer perceptron owned by block i, with two hidden layers of 16 tanh units. Its inputs are the block-local coordinates (ξ, η) ∈ [−1, 1]² and τ = (t̃, sin 2πt/24 h, cos 2πt/24 h). The last layer starts at zero, so every reconstruction starts at the background. Evaluation at a point uses the network of the block that contains the point. Nothing is blended.

The scales s_c and s_s are the bias-like and structured standard deviations of the background error (Section 3.8), so a and f are dimensionless with unit prior variance.

3.6.3 Observation operators

An observation is y_k = H_k(x) + ϵ_k with ϵ_k ∼ 𝒩(0, σ_k²). Because the fields are functions of continuous coordinates, a station operator evaluates the field at the station’s own position and time and needs no interpolation from a grid:

H_k^W(x) = W(λ_k, φ_k, t_k),    H_k^Q(x) = Q(λ_k, φ_k, t_k).    (3.14)

W-type observations stand for a column vapour measurement at a surface site (a GNSS receiver, or a sounding). Q-type observations stand for the transport computed from a radiosonde profile and are the only direct information on the rotational part of Q. Gridded rainfall and evaporation are not observations of the state. They enter the ledger as the forcing S_iⁿ, with their error carried by σ_m.

3.6.4 Objective

The maximum a posteriori estimate under Gaussian errors minimises the weak-constraint objective (Trémolet 2006)

$$J\left( \mathbf{a},\theta \right) = \underset{J_{o}}{\underbrace{\sum_{k}^{}\frac{(y_{k} - H_{k}(x))^{2}}{\sigma_{k}^{2}}}} + \underset{J_{b}}{\underbrace{\lambda_{b}\sum_{k}^{}\mathbf{f}^{(k)\top}(\mathsf{C}^{(k)} + \varepsilon\mathsf{I})^{- 1}\mathbf{f}^{(k)} + \parallel \mathbf{a} \parallel^{2}}} + \underset{J_{m}}{\underbrace{\lambda_{r}\sum_{i,n}^{}(\frac{R_{i}^{n}}{\sigma_{m,i}^{n}})^{2}}} + \underset{\text{interface}}{\underbrace{\lambda_{\Gamma}J_{\Gamma}}}.\quad\quad\text{(3.15)}$$

Background term. f^(k) is the network output for component k on the quadrature lattice of cell centres and window times. C^(k) = C_t ⊗ C_φ ⊗ C_λ is a separable Gaussian correlation matrix with length L^(k) and time T_c^(k), and ε = 10⁻² is a nugget. The quadratic form is evaluated in the eigenbasis of the three factors. For the block variants C is built block by block, so the prior shares nothing across faces. A first version with a white penalty on the network output let the block fields fit station noise exactly, and a first-order Sobolev penalty did the same more slowly. Both are recorded in Chapter 4 as rejected designs.

Ledger term. R_iⁿ is (3.10) evaluated on the learned fields, with the forcing an analyst would have. Its error model is heteroscedastic,

(σ_m, iⁿ)² = σ₀² + (σ_p P̄_iⁿ)²,    (3.16)

where P̄_iⁿ is the block-window mean rain rate of the product. The floor σ₀ is the closure error of a reference analysis in dry conditions and σ_p expresses that the budget closes worse where it rains.

Weights. λ_r = 1 throughout, because σ_m is calibrated. λ_b and λ_Γ are tuned on development events with the same search budget for every variant.

3.7 Coupling variants

The variants differ only in the transfer that a block books through a face it shares with a neighbour. Let Q_n⁽ⁱ⁾ and Q_n^(j) be the normal transport on a shared face evaluated from the field of block i (west or south) and block j (east or north).

Table 3.2. Coupling variants. ⟨⋅⟩_Γ is the mean over interface quadrature points and hours, δ_W and δ_Q are tolerance scales.

Under C3 the two blocks book the same number with opposite signs, so Proposition 1 applies to the fitted fields: the sum of the block residuals equals the domain residual to rounding error. Under C0, C1 and C2 it does not, and the difference is the interface defect of Proposition 8. C3 is an accounting identity. Whether the shared transfer is accurate is a separate, measured question.

3.8 Error statistics

The evidential weight of every term sits in its error statistics. The background error of component k is modelled as a constant per event plus a structured field,

$$x_{b}^{(k)} - x^{\star (k)} = c^{(k)} + e^{(k)}(\lambda,\varphi,t),\quad c^{(k)} \sim \mathcal{N}\left( 0,s_{c}^{(k)2} \right),\quad Cov(e,e\prime) = s_{s}^{(k)2}\exp\left\lbrack - \frac{d^{2}}{2L^{(k)2}} - \frac{(t - t\prime)^{2}}{2T_{c}^{(k)2}} \right\rbrack.\quad\quad\text{(3.17)}$$

s_c is the root-mean-square of the event-mean error and s_s the root-mean-square of the remainder. L and T_c come from lagged autocorrelations. σ₀ and σ_p in (3.16) are fitted by least squares of the squared ledger residual of the reference on (1, P̄²). In a simulation test the truth is known on development events, so all of these are estimated there and frozen. With real observations, s_c, s_s, L and T_c come from station-minus-background statistics over a training period, σ_k from instrument specifications plus a representation term, and σ₀ and σ_p from the closure of the background product itself.

3.9 Baselines in the same notation

B0 is x_b.

B1, optimal interpolation. With innovation vector d = y − H(x_b), kernel k(⋅, ⋅) = s_c² + s_s²exp [⋅] from (3.17) and R = diag(σ_k²), the increment at any point is k_*^⊤(K + R)⁻¹d. No physics enters.

B3, weak-constraint variational smoother on the grid. The control vector is a white field χ and three constants c. The increment is s_cc + s_s G_TG_Lχ, where the G are Gaussian filters of standard deviation $L/\sqrt{2}$ and $T_{c}/\sqrt{2}$, so the implied covariance is (3.17). The objective is  ∥ χ∥²+ ∥ c∥² + J_o + J_m with the same ledger, the same σ_m and transports on faces taken as the mean of the two adjacent nodes. B3 differs from D-LEGOS only in how the state is represented, which is the comparison a data-assimilation reader asks for first.

B2, single neural field. One network over the whole domain with the same number of trainable parameters as all the block networks together, the same baseplate, the same ledger, and the Gaussian prior applied over the whole domain.

3.10 Optimisation and uncertainty

J is deterministic given the observations, so it is minimised by L-BFGS with a strong Wolfe line search. The baseplate is fitted first with the networks at zero, a linear least-squares problem in twelve numbers, then all parameters together. Every fit reports the number of function evaluations and the relative decrease of J over the last twenty, and a subset is rerun at three times the iteration budget to measure how far the scores move.

Uncertainty is represented by randomised maximum likelihood. Member m minimises J with observations perturbed by draws from R and, in the discrepancy-aware variant, with the background perturbed by a draw from (3.17) and the rainfall forcing by a new draw of its error. In a linear Gaussian problem the members are exact posterior samples. Omitting the background and forcing perturbations gives the fixed variant, which treats the driver as known and is the construction Proposition 5 warns about.

3.11 The rainfall-allocation sub-problem

Hypothesis H1 concerns a narrower task: allocate a coarse rainfall total d_c among the fine cells j ∈ c. With features z_j (elevation, position within the parent, month, neighbouring parents), a climatological share r_j, m = p̄_j, m/C_c[p̄_⋅, m] for calendar month m and a network score g_θ, the estimators compared in Chapter 6 are

$${\widehat{p}}_{j}^{\text{free}} = softplus\left( g_{\theta}\left( \mathbf{z}_{j},d \right) \right),\quad\quad{\widehat{p}}_{j}^{\text{hard}} = d_{c}\,\frac{A_{c}\, e^{g_{\theta}\left( \mathbf{z}_{j},d \right)}}{\sum_{l \in c}^{}a_{l}\, e^{g_{\theta}\left( \mathbf{z}_{l},d \right)}},\quad\quad{\widehat{p}}_{j}^{\text{dep}} = d_{c}\,\frac{A_{c}\, r_{j,m}\, e^{g_{\theta}\left( \mathbf{z}_{j},d \right)}}{\sum_{l \in c}^{}a_{l}\, r_{l,m}\, e^{g_{\theta}\left( \mathbf{z}_{l},d \right)}}.\quad\quad\text{(3.18)}$$

The hard forms satisfy C_c[p̂] = d_c by construction, in the manner of the constraint layers of Harder et al. (2023). The departure form starts at the seasonal allocation when g_θ = 0 and learns only what the season does not explain. Proposition 7 applies to both hard forms and to the seasonal allocation itself.

3.12 The pressure-level class

The full class takes (u, v, T, q) on pressure levels as unknowns, with one block field of four outputs per level. Φ is diagnosed from T_v by the hydrostatic equation with a boundary value from the background. ω is taken from the background in a first comparison and later diagnosed from continuity. F_h, Q̇ and S_q are prescribed from a reference or bounded by a prior and are never free functions, so that a wrong closure appears as reconstruction error. Levels below the surface are masked. The objective adds the non-dimensionalised residuals of (3.1) and (3.2) at collocation points to (3.15), and the ledger is (3.10) with

$$W = \frac{1}{g}\sum_{l}^{}{\bar{q}}_{l}\,\Delta p_{l},\quad\quad\mathbf{Q} = \frac{1}{g}\sum_{l}^{}{\overline{q\mathbf{v}}}_{l}\,\Delta p_{l}\quad\quad\text{(3.19)}$$

evaluated by the layer quadrature of Appendix A, which cuts the lowest layer at p_s. Surface stations then observe the lowest unmasked level and soundings observe profiles, which is where the vertical-information factor of H2 acts in full. The code accepts this class through the same interfaces. It has not been run, because hourly pressure-level fields for the event set are two orders of magnitude larger than the column fields, and I state the column result as a result about the column form only.

3.13 Assumptions and limits of the formulation

1. Hydrostatic, mesoscale. Nothing here represents convective cells.
2. Vapour only. Condensate storage and transport are in the ledger error, not in the state.
3. Gaussian error models with fixed, calibrated statistics. Heavy-tailed rainfall error is represented only through the proportional term of (3.16).
4. The baseplate is linear in space and time. A background whose bias has structure at the scale of a few blocks is not captured by it.
5. Block cores are rectangles. Terrain-aligned cores are a planned variant and are not part of the executed test.
6. Propositions 1 to 9 are statements about stated graphs and operators. None of them proves positivity of humidity, stable integration, identifiable transfers or calibrated uncertainty. Those are measured.

Table 3.4. Symbols and their names in the code (package dlegos).

Chapter 4. Research design and methods

4.1 The rule

Every claimed improvement in this thesis is measured on information the model did not use, against a baseline given the same data and the same computational budget. The design below exists to make that rule enforceable.

The work is staged. Stage 0 is a set of numerical unit tests of the ledger with no learning. Stage 1 is a simulation test in which truth is known. Stage 2 is the reconstruction of observed events over Jamaica and needs station data that this thesis does not yet hold; its protocol is Chapter 8. A second, separate experiment tests hypothesis H1 on gridded rainfall for three islands (Section 4.11).

4.2 Domain, grid and blocks

The analysis domain is 14 to 22°N, 82 to 72°W: Jamaica, eastern Cuba, western Hispaniola, the Cayman Islands and the sea across which the low-level jet carries its moisture. The grid is that of the nature run, 0.25° in latitude and longitude, 32 by 40 cells. Blocks are 8 by 8 cells (two degrees), which gives a lattice of 4 by 5 blocks with 31 internal faces. Ledger windows are three hours. Event windows are 48 hours from 00 UTC.

4.3 Data

Table 4.1. Data used in the Stage 1 simulation test. All are public and were retrieved on 11 November 2026.

The nature run and the background come from different models, different assimilation systems and grids that differ by a factor of ten. A simulation test whose nature run and background share a model flatters the method under test (Hoffman and Atlas 2016); this one does not have that defect, although both products assimilate many of the same observations.

The background column vapour and transport are computed here from the pressure-level fields by layer quadrature from the surface pressure to 300 hPa (Section 3.12), then interpolated bilinearly in space and linearly in time to the analysis grid and to hourly steps. The background is therefore smooth, coarse and biased: on the development events its column vapour is 0.7 to 6.1 kg m⁻² too dry in nine of ten events, and its transport error is 42 to 125 kg m⁻¹ s⁻¹ outside cyclones and 350 to 550 inside them.

4.4 Events

Twenty-five events were chosen by rule from two periods: ten development events from 2016 to 2019, used for every tuning decision, and fifteen test events from 2020 to 2025, fitted once after the plan was frozen. Selection used the background reanalysis and the best tracks only, never the nature run. Five regimes are represented, following the event classes of the original design.

Table 4.0. Event catalogue. Score: best-track wind in knots (TC), domain-mean rain in mm per day (EARLY, LATE, DRY), 925 hPa zonal wind in m s⁻¹ in the low-level-jet box (MSD).

Two test cyclones, Beryl and Melissa, were named in advance because the thesis proposal names them. The third test cyclone and every other event were chosen by score. Two late-season test events sit close to tropical systems that the rule does not count. During the window of 15 November 2020 Hurricane Iota tracked along 13°N, south of the domain edge at 14°N. During the window of 16 November 2023 an unnamed system, carried in the best track as a disturbance and never classified as a tropical cyclone, crossed the domain. The rule looks only at best-track tropical-cyclone fixes inside the domain, and I have left the catalogue as the rule produced it.

4.5 Synthetic observations

Observations are drawn from the nature run at the coordinates of the real network (Table 2.1).

• Column vapour at each surface site at the eight synoptic hours, with independent Gaussian error of standard deviation 2 kg m⁻².
• Soundings at Kingston and Grand Cayman at 00 and 12 UTC: the transport vector with error 30 kg m⁻¹ s⁻¹ per component, and the column vapour.
• Rainfall product: the true hourly rainfall multiplied by a log-normal error with log standard deviation 0.5, correlated over three grid cells, with an optional bias factor (1 + α) used in the stress test.
• Evaporation: the background’s, from its latent heat flux.

The network is thinned to 100, 50, 25 and 10 % of the surface sites (23, 12, 6 and 2 stations), in nested subsets, with and without the soundings.

Three of these choices are assumptions of the simulation and not facts about the real network: that each surface site yields a column-vapour value of GNSS quality, the size of the sounding transport error, and the rainfall error model. Each is a single stated number so that it can be replaced when the Jamaican instruments are audited.

4.6 Error statistics

All error statistics were estimated on the development events and frozen (Section 3.8).

Table 4.2. Calibrated error statistics.

The background statistics exclude the cyclone events, whose second moments are an order of magnitude larger and would otherwise set the scales for every regime. Ledger values are in kg m⁻² h⁻¹ and were computed with the forcing an analyst would have: the noisy rainfall product and the background evaporation.

4.7 Methods compared and matched resources

Table 4.3. Methods. The learned variants share the baseplate, the prior, the ledger and the optimiser, and differ only as shown.

Each block network has 419 parameters and there are twenty. The single field B2 has the hidden width that brings its count closest to theirs. All learned variants use the same optimiser and the same iteration budget (40 L-BFGS iterations for the baseplate, then 300 for all parameters). Wall time is reported for every fit, because matched parameters do not mean matched cost: B2 evaluates a network twenty times larger at every quadrature point.

4.8 What the development runs changed

The design of the earlier research-design draft was not run as written. Six departures were made on development events before the plan was frozen, and I list them because each is a decision a reader could dispute.

1. Column form. The executed state is (W, Q), the vertical integral of the pressure-level class. Hourly pressure-level fields for the event set were out of reach of the available storage.
2. Windows of 48 hours in place of 72, to bound computing time.
3. A baseplate. Without a shared coarse component, block fields left the background’s domain-wide dry bias uncorrected in the eleven blocks with no station. Twelve shared parameters removed it.
4. The prior. A white penalty on network output let the block networks fit station noise exactly while the objective kept falling. A first-order Sobolev penalty did the same more slowly. The Gaussian-covariance form of Eq. (3.15), built from the calibrated scales, gave observation misfits close to their expected value and stable scores when the iteration budget was tripled.
5. Ledger at block scale only. The cell-scale budget of the nature run closes too poorly to constrain anything (Chapter 5), so the ledger is applied to blocks and windows, as the formulation intends.
6. The grid variational smoother as the conventional comparator (B3), in place of an ensemble Kalman smoother. It shares the covariance and the ledger with the learned variants, which isolates the parameterisation.

4.9 Tuning

Each learned variant was tuned on five development events (one per regime) at the 100 % and 25 % networks, on the same grid: λ_b ∈ {0.3, 1, 3} for C0, C3 and B2, and λ_Γ ∈ {0.1, 1, 10} for C1, C2 and C4 at the λ_b selected for C0. The score was the mean of the two primary outcomes relative to the unadjusted background. Selected values: λ_b = 1 for C0, C1, C2, C4 and B2; λ_b = 0.3 for C3; λ_Γ = 0.1 for C1, C2 and C4. B1 and B3 have no tuned weights.

4.10 The frozen plan

The plan was written to configs/plan.json at 09:02 UTC on 11 November 2026, with SHA-256 hashes of every source file, of the calibration and of the event catalogue, before any test event was fitted.

Primary outcomes. Root-mean-square error against the nature run of (i) the transport vector Q over all cells and hours, and (ii) the transport divergence averaged over ledger control volumes (two-degree blocks, three hours). Both are expressed relative to the error of the unadjusted background for the same event, so that events of very different magnitude can be pooled.

H2 decision rule. Supported if, for both primary outcomes, the paired difference C3 minus C1 is negative with a 95 % interval excluding zero at the 25 % and 10 % networks, and does not favour C1 at the full network by more than the minimum detectable effect. Partially supported if this holds for one outcome. Otherwise not supported.

Minimum detectable effect. From the spread of the paired difference on development runs, for fifteen events at 80 % power: 0.020 for transport and 0.048 for block divergence, in units of the background error.

H3 decision rule. Supported if the discrepancy-aware ensemble has lower CRPS than the fixed ensemble for withheld column vapour and for transport, with intervals excluding zero, and its 90 % interval coverage lies between 80 and 98 %. Wider intervals alone do not count.

H1 decision rule. Supported if the hard seasonal-departure model has test error no higher than its matched free counterpart at zero driver bias, and the paired difference grows with the size of the bias. The crossover is estimated from the sweep.

Secondary analyses, declared in advance and reported as such: the effect of the ledger (each method against its no-ledger control); the parameterisation (C3 against B3); soundings removed at the 25 % network; a rainfall-product bias of ±25 and ±50 %; no baseplate; one-degree blocks; six-hour windows; and a convergence audit at three times the iteration budget.

Inference. Intervals are percentile intervals from 10,000 bootstrap resamples of events (Künsch 1989). The event is the independent unit. No correction for multiple comparisons is applied to the secondary analyses, and none of them is used to support a hypothesis.

4.11 Verification

All scores are computed from the assembled gridded fields. Divergence and curl at grid scale are computed with centred differences of the spherical operators, a discretisation the fitting never uses, and block-scale divergence is the area mean of the finite-volume divergence taken after assembly. Interface jumps, ledger residuals and the cancellation defect of Proposition 8 are reported for every learned fit. Ensemble scores are the fair CRPS (Ferro 2014), the coverage and width of 90 % intervals, and the energy and variogram scores (Gneiting and Raftery 2007; Scheuerer and Hamill 2015) for the vector of twenty block-mean divergences.

4.12 The allocation experiment (H1)

Data. CHIRPS v2.0 monthly rainfall at 0.05° (Funk et al. 2015) for Jamaica, Hispaniola and Puerto Rico, 1981 to 2016; ETOPO2v2 relief averaged to the CHIRPS cells; and the GPCC Full Data Monthly product at 0.25° (Schneider et al. 2018), a gauge-only analysis, as an independent coarse driver. Each 0.25° parent holds 25 fine cells, and parents with fewer than five land cells are dropped.

Split. Training 1981 to 2005, validation 2006 to 2010, test 2011 to 2016. Climatological shares come from the training years only.

Estimators. Uniform allocation; seasonal allocation by climatological share; a free network; a hard-constrained network; a hard-constrained network that predicts departures from the seasonal share; and its matched free counterpart (Eq. 3.18). The networks have two hidden layers of 32 units and 1,633 parameters. Their inputs are elevation, elevation relative to the parent mean, position within the parent, the month, and the ratio of the driver to its climatology in the parent and its eight neighbours. Latitude and longitude are not inputs, so a model can be applied to an island it was not trained on.

Experiments. (i) Exact coarsening: the driver is the parent mean of the fine truth. (ii) Controlled bias: the fitted models are held fixed and the driver is multiplied by (1 + α) for α from −0.5 to +0.5. (iii) Independent driver: the driver is GPCC, with models trained on exact coarsening (the perfect framework) and retrained on GPCC (the imperfect framework), following the comparison of Meer, Roda Husman, and Lhermitte (2023). (iv) Transfer: train on two islands, test on the third.

Inference. Paired differences with a bootstrap over the six test years.

4.13 Reproducibility and claim checks

Every fitted model writes one line to a results file with its scores, diagnostics, optimiser record and wall time. Tables in Chapters 5 and 6 are generated from those files. Every number quoted in the prose is re-emitted from the same files by a declarative claim check, with a negative control in which three planted errors must be caught and no others (Appendix D).

4.14 Threats to validity

Table 4.4. Threats to validity and the response to each.

Chapter 5. Results of the simulation test

The plan of Chapter 4 was frozen at 2026-11-11T09:02:05Z UTC and the fifteen test events were then run once: 660 reconstructions in the main experiment (11 methods at four network densities), followed by the stress tests and ensembles of Sections 5.5 and 5.6. This chapter reports what came out. Every error is a root-mean-square error against the nature run, divided by the same error for the background, so 1 means no better than the first guess. Intervals are 95 % bootstrap intervals over events. Divergence and curl are recomputed from the assembled fields with a discretisation that the fitting never used.

Averaged over the test events the background’s errors are 5.39 kg m⁻² in column vapour, 174 kg m⁻¹ s⁻¹ in transport and 0.94 kg m⁻² h⁻¹ in block-scale divergence, with a domain-mean dry bias of −2.35 kg m⁻².

5.1 The closure floor: how much the budget can tell us

Before any reconstruction, the ledger was evaluated on the nature run itself, with its own rainfall and evaporation, and on the background with the same forcing. If ERA5 does not close its own budget better than the background does, the budget cannot distinguish a good field from a poor one at that scale.

Table 5.1. Root-mean-square ledger residual of ERA5 divided by that of the background, with true evaporation minus rainfall, by control-volume size and window.

Figure 5.1. The closure floor. Below the dashed line ERA5 closes its own column budget better than the background does.

At the grid scale and one hour, ERA5’s own residual is larger than the background’s (ratio 1.10 on the test events). An hourly budget on 0.25° cells carries no information here, for the reasons of Section 3.3: condensate is neglected and the analysis increments act as sources. The ratio falls as the control volume grows. At 2° and 3 h, the setting used throughout, it is 0.73 on the test events and 0.62 on the development events.

Two consequences follow. A perfect reconstruction could lower the background’s ledger residual by 27 % on the test events, and by 11 % in the early wet season. And any reconstruction whose ledger residual is smaller than ERA5’s own, 0.59 kg m⁻² h⁻¹ at this size, has closed the budget more tightly than the truth does.

5.2 The main experiment

Table 5.2. Errors relative to the background, full surface network (23 sites), 15 test events.

Table 5.3. As Table 5.2, 25 % network (6 sites).

Table 5.4. As Table 5.2, 50 % network (12 sites).

Table 5.5. As Table 5.2, 10 % network (2 sites).

Figure 5.2. Errors relative to the background against network density, with 95 % intervals over events. B1 uses no budget; the other four use the same weighted ledger.

5.2.1 Block-scale divergence: the ledger is what matters

Every method that uses the ledger lowers the error in block-scale divergence. Leaving aside the uncoupled blocks (C0, at 0.91; Section 5.3), they reach between 0.81 and 0.83 of the background’s at the full network, and to between 0.85 and 0.87 with two surface sites. Every method that does not use it ends above the background, between 1.06 and 1.14.

The last of those sentences deserves attention. Statistical interpolation of sparse observations makes the block-scale divergence worse than leaving the background alone (B1: 1.106 (1.049 to 1.172)). Two sounding sites pull the transport towards the truth locally, and a local correction to a transport field is a dipole of spurious divergence. The ledger removes this damage and then some. Paired against its own no-ledger control, the variational smoother gains −0.306 (−0.432 to −0.200) and the shared-flux blocks gain −0.243 (−0.316 to −0.179), in units of the background error. The effect is the same size at every network density, because it does not come from the stations.

5.2.2 Transport: nothing recovers it

No method lowers the transport error by more than 2.5 % at the full network. All ten sit between 0.975 and 0.985, and most intervals include 1. The ledger’s own contribution is −0.008 (−0.012 to −0.004) for the shared-flux blocks and −0.001 (−0.011 to 0.007) for the smoother at the 25 % network, both well under the smallest effect of interest fixed in advance (0.0196).

This is Proposition 4 at work. A budget constrains the divergent part of the transport, and in the trades the transport is almost entirely rotational. The error in grid-scale curl is unchanged by every method, between 0.985 and 1.018 of the background’s. Two soundings cannot fix what the budget cannot see.

5.2.3 Column vapour: the stations help, the networks do not, and geometry beats density

Statistical interpolation and the variational smoother lower the column-vapour error to 0.776 and 0.795 of the background’s with the full network. The neural variants do worse, between 0.85 and 0.90, and the shared-flux blocks are behind the smoother by 0.056 (0.009 to 0.103).

The dependence on density is not monotone, and that was not expected. With 12 sites every neural variant is worse than with 6, and several are worse than the background: the single neural field at 1.061 and the state-matched blocks at 1.023. The shared-flux blocks and the single field are each worse than the background in 9 of 15 events at that density.

The residual bias explains it (Table 5.6). Most of the gain in column vapour is the removal of the background’s domain-wide dry bias. How well that bias is removed depends on which sites are kept. The stations sit where the background is least wrong. Averaged over the test events the background’s bias in column vapour is −2.36 kg m⁻² over the domain and −3.03 south of 18°N, but only −0.64 at the 23 sites, of which 16 lie north of 19.8°N and 1 lies south of 18°N. The six sites of the 25 % network see a bias of −0.94; the six added to make the 50 % network see −0.15, so the denser network reports that the background is nearly unbiased, and the fitted offset is carried over open water that no station observes. In blocks with no station the neural variants are above the background’s error at the two denser networks.

Table 5.6. Domain-mean bias in column vapour after reconstruction (kg m⁻²; background −2.35) and, in brackets, the error relative to the background in blocks with no station.

The density experiment used one nested thinning of the network. It therefore cannot separate the number of sites from their placement, and Table 5.6 says that placement dominates. This is a limitation of the design (Section 9.5) and a finding about the problem: a low-order bias model fitted to stations clustered on land extrapolates over water, and the neural variants, which give that model more freedom, extrapolate worse than interpolation on a grid.

5.3 Hypothesis H2: shared flux against the alternatives

H2 said that coupling the blocks through one shared signed flux per face (C3) would beat matching the states across faces (C1) on the two primary outcomes when observations are sparse.

Table 5.7. Paired differences in relative error, shared flux minus the alternative, with 95 % intervals over 15 events. Negative favours shared flux. Smallest effects of interest: 0.0196 for transport and 0.0485 for block-scale divergence.

Figure 5.3. Paired differences between shared flux and each alternative. The grey band is the smallest effect of interest fixed before the test.

H2 is not supported. Against state matching, the difference in transport error is −0.001 (−0.017 to 0.019) at the 25 % network and 0.007 (−0.011 to 0.027) at 10 %; in block-scale divergence it is −0.013 (−0.042 to 0.013) and −0.012 (−0.043 to 0.013). Every interval includes zero and every point estimate is well inside the smallest effect of interest. The same holds against penalised flux, against a single neural field of matched size and against the conventional smoother. How the field is represented, and how the blocks are joined, made no measurable difference to either primary outcome once the ledger was in the objective.

One disclosure. I tabulated the comparison when six events had finished, to check that the analysis script ran. At that point the divergence difference was about −0.05 with an interval touching zero. With all fifteen events it is −0.01. No setting was changed between the two looks, and the interim figure is reported here only so that the record is complete.

Coupling of some kind does matter. The paired difference between shared-flux blocks and blocks with no coupling at all (C0) is −0.110 (−0.162 to −0.068) in block-scale divergence. An uncoupled block has no information about what its neighbours are doing with the same water.

5.3.1 Exact cancellation did not buy accuracy

The shared-flux variant does what the propositions of Chapter 3 say. Interior transfers cancel to rounding error (median defect 5.1e−16), as they do for the single field (4.5e−16). With state matching or a flux penalty they do not: a quarter of the gross internal transfer is left over as spurious sources (median defect 0.28 for C1 and 0.25 for C2), and half for uncoupled blocks (0.51).

Figure 5.4. Left: the fraction of interior transfer that fails to cancel. Right: the mean jump in transport across block faces.

Yet C1, with a quarter of its internal bookkeeping wrong, has the same block-scale divergence error as C3. The reason is that the verification does not use the method’s own bookkeeping. It differences the assembled field, and what matters there is the field, not whether two blocks agreed on a number for their common face.

5.3.2 Shared flux without continuity damages the grid scale

Figure 5.5. Block-scale against grid-scale divergence error, 25 % network. Hollow markers are the no-ledger controls.

At the grid scale the picture changes. The shared-flux blocks have a divergence error of 1.28 times the background’s, a quarter worse than doing nothing, and so do the uncoupled blocks (1.24). Sharing the flux through a face does not make the state continuous across it: the mean jump in transport at block faces is 34 kg m⁻¹ s⁻¹ for C3 against 9 for C1. A centred difference across a face turns each jump into a line of spurious divergence. The variants that match the state (C1, C2) stay near 0.99, and the smoother and the single field, which have no faces, reach 0.95 and 0.95.

The exploratory variant C4, which shares the flux and matches the state, has the lowest jump (7), exact cancellation, a grid-scale error of 0.97 and a block-scale error of 0.829. It is better than C1 in block-scale divergence by −0.013 (−0.026 to −0.001) at the full network. The difference is a quarter of the smallest effect of interest and C4 was not a registered hypothesis, so this is a lead and not a result.

5.3.3 Closing the budget more tightly than the truth

The shared-flux fields close the ledger, computed with the noisy rainfall product they were given, to 0.35 kg m⁻² h⁻¹. ERA5 closes its own, with its own rainfall, to 0.59. The two are not like for like, but the direction is clear: the fit is tighter than the truth. A reconstruction in that position has moved part of the rainfall product’s error into the transport (Proposition 5). The stress test of Section 5.5 measures what that costs when the product is biased.

5.4 By regime

Table 5.8. Errors relative to the background by regime, 25 % network. Three events per regime.

Figure 5.6. Errors relative to the background by regime, 25 % network.

The ledger earns most where it rains most. In the cyclone events the smoother and the shared-flux blocks reach 0.76 and 0.75 of the background’s block-scale divergence error, and in the late wet season 0.77 and 0.78. In the midsummer events, when the jet is strong and little rain falls, they reach only 0.95 and 0.93. Rainfall is the term that ties the budget to the divergence; without it the ledger has little to say.

In the early wet season every method in the table lowers the transport error, by between 5 and 11 %. The largest gain in any other regime is 3.7 %. In the cyclone events four of the five methods in the table make the transport slightly worse than the background. With three events per regime these are descriptions and not tests.

5.5 Stress tests and ablations

All of these use the 25 % network and all 15 test events.

5.5.1 Without soundings

Table 5.9. Errors relative to the background with the two sounding sites removed, beside the same method with soundings (in brackets).

Removing the soundings separates two effects that the main experiment mixes. The small gain in transport came from the soundings: without them every method is within 1 % of the background. The gain in block-scale divergence did not: the ledger methods are at 0.83 to 0.86 with or without them. And the damage that interpolation does to divergence came from the soundings alone: with none, B1 leaves the transport untouched and its divergence error is that of the background.

The ledger therefore improves the divergence of a transport field that it has no direct observation of, from rainfall and column vapour alone. That is the property the method was designed to have.

5.5.2 A biased rainfall product

The rainfall product was multiplied by (1 + α) before the reconstruction saw it.

Table 5.10. Block-scale divergence error relative to the background when the rainfall product is biased, with 95 % intervals. The no-ledger controls do not use rainfall: 1.174 for the smoother and 1.066 for the shared-flux blocks.

Figure 5.7. Divergence and transport errors against the bias in the rainfall product. Dashed lines are the no-ledger controls.

The weighted ledger tolerates a rainfall bias better than the allocation result of Chapter 6 might suggest. A product that reports half the true rain costs the shared-flux blocks 44 % of their gain over the background and the smoother 50 %, and both remain far below their no-ledger controls. A bias of 25 % in either direction costs at most 17 %.

The response is not symmetric. Too little rain hurts more than too much. A likely reason is in the tolerance, which grows with the reported rainfall: an inflated product loosens the constraint where it is wrong, and a deflated product tightens it there. I did not test this explanation.

The contrast with Chapter 6 is the point. There the constraint was exact and the driver’s bias passed through in full. Here the constraint is weighted, the weight falls where rain is heavy, and half the benefit survives a 50 % error. The transport error does not respond to the rainfall bias at all, as expected for a quantity the ledger does not control.

5.5.3 Ablations

Table 5.11. Ablations of the shared-flux blocks (C3), 25 % network, relative to the background. Block-scale divergence for the block and window ablations is verified on their own control volume, with the background verified on the same one.

• The shared basis carries the column-vapour gain and nothing else. Without it the column-vapour error rises from 0.82 to 0.92 and the divergence error does not move. This agrees with Section 5.2.3: the gain in column vapour is the removal of a large-scale bias.
• Smaller blocks work, a little less well, as the closure floor of Section 5.1 leads one to expect. At 1° the divergence error is 0.85 of the background’s at that scale.
• A longer window gives the same divergence result and a worse transport.
• Three times the iterations changes nothing for the better. Column vapour is unchanged, and transport and divergence are slightly worse. The fixed budget of 300 iterations was not holding the method back.

5.6 Hypothesis H3: uncertainty

H3 said that an ensemble which resamples the background and the rainfall product as well as the observations would be better calibrated than one that perturbs the observations alone. Two ensembles of 8 shared-flux reconstructions were run for each test event at the 25 % network: “fixed”, in which only the observations are perturbed, and “aware”, in which each member also draws a background perturbation from the calibrated error covariance and a new realisation of the rainfall product. They are scored on column vapour at the surface sites that were withheld from the 25 % network, and on the transport field.

Table 5.12. Ensemble scores, means over 15 test events. Differences are aware minus fixed with 95 % intervals over events.

Figure 5.10. Coverage, spread and error of the two ensembles.

An ensemble that perturbs only the observations is badly over-confident. Its 90 % interval contains the truth 20 % of the time for column vapour and 9 % for transport, because its spread (0.74 kg m⁻²) is a fraction of its error (4.00). Nearly all of the error is in the background, and that ensemble holds the background fixed.

Resampling the background and the rainfall raises coverage to 75 % and 75 % and lowers the CRPS of both variables. For column vapour the spread now matches the error of the mean (ratio 1.05). For transport it is 80 % of the error, so the ensemble remains under-dispersed there.

The frozen rule asked for two things: a lower CRPS for both variables with intervals excluding zero, and coverage between 80 and 98 %. The first is met. The second is not, for either variable. By the rule, H3 is not supported. The plain reading is that resampling the inputs is necessary and, with the background-error model used here, not sufficient: the Gaussian, stationary covariance of Section 3.8 under-states the background’s transport error in the events that matter.

Two cautions. The error of the ensemble mean is higher for the aware ensemble, because each of its eight members starts from a perturbed background and eight is too few for the perturbations to average out. And with eight members a 90 % interval is estimated from the extremes of a very small sample, so the coverage figures are themselves coarse.

5.7 Two cyclones

Figure 5.8. Hurricane Beryl, window starting 3 July 2024, 25 % network. Top: column vapour at hour 24 for the nature run, and errors of the background, the smoother and the shared-flux blocks. Bottom: 48-hour mean block-scale divergence and its errors.

Figure 5.9. As Figure 5.8 for Hurricane Melissa, window starting 27 October 2025.

The two named cyclones show the general result in one picture each. The background places the storm’s moisture too far from the centre and has the wrong sign of block-scale divergence under it. Both ledger methods remove a large part of the divergence error in the blocks where the rain fell. Neither changes the pattern of the column-vapour error by much, and the block edges are visible in the shared-flux field as the discontinuities of Section 5.3.2.

5.8 Cost

The block decomposition is 6.2 times faster than a single network of the same size, because each block’s network sees only its own points. That is a real advantage of decomposition, and it is one of speed. It is also 2.3 times faster than the smoother as implemented here, a comparison that says more about my implementation of the smoother than about the methods.

5.9 Summary of findings

1. The budget carries information only at coarse scales. ERA5 closes its own column budget better than the background at 2° and 3 h (ratio 0.73) and worse at 0.25° and 1 h (1.10).
2. The ledger is the active ingredient. With it, block-scale divergence error falls to 0.81 to 0.83 of the background’s (uncoupled blocks: 0.91). Without it, assimilating the same observations leaves it at 1.06 to 1.14.
3. The representation is not. A grid, one neural field and four ways of coupling neural blocks give the same primary outcomes within the smallest effect of interest. H2 is not supported.
4. Transport is not recovered by any method, by more than 2.5 %. The rotational part is invisible to a budget.
5. Column vapour gains are bias removal, depend on where the stations are more than on how many there are, and are larger for the conventional methods than for the neural ones.
6. Shared flux gives exact cancellation and discontinuous fields. Exact cancellation did not improve accuracy; the discontinuities raise grid-scale divergence error to 1.28 of the background’s. Adding state matching (C4) removes them.
7. The weighted ledger degrades gracefully under a biased rainfall product, losing about half its gain at a 50 % under-estimate.
8. Decomposition is faster, by a factor of 6 against a single network.

Chapter 6. Rainfall allocation under driver error: results for three islands

This chapter tests hypothesis H1 on monthly rainfall over Jamaica, Hispaniola and Puerto Rico. The task is narrow: given a rainfall total on a 0.25° parent cell, share it among the 0.05° land cells inside. The design is in Section 4.12 and the estimators in Section 3.11. All scores are root-mean-square errors in mm per month over the land cells of the test years 2011 to 2016, against CHIRPS.

The sample is 24 parents for Jamaica, 147 for Hispaniola and 21 for Puerto Rico, with 459, 3,000 and 380 land cells. Hispaniola supplies three quarters of the parents, so every pooled figure is mostly Hispaniola and the island columns should be read beside it.

6.1 With an exact driver, a seasonal lookup is the estimator to beat

When the driver is the parent mean of the fine field itself, the only error is in the allocation. Table 6.1 and Figure 6.1 give the result.

Table 6.1. Allocation error with an exact driver (mm per month). Learned estimators are means of three seeds.

Figure 6.1. With an exact driver, the seasonal share removes two thirds of the uniform-allocation error and the networks that ignore it barely beat uniform.

Four things follow.

The seasonal share does most of the work. A table of climatological shares by calendar month, with nothing fitted, removes 68 % of the uniform-allocation error over the three islands (68 % for Jamaica). Where rain falls within a parent is mostly a fixed function of terrain and season.

A network that is not given the seasonal share does not find it. The free network and the hard-constrained network, which see elevation, position and month but not the share, are within 10 % of uniform allocation. With 1,633 parameters and no coordinates they cannot represent a cell-by-cell climatology, and that is by design: a model that memorised the map could not be moved to another island.

The hard constraint helps, in both forms. The hard-constrained network is 6.1 % below its free twin (paired difference −1.68 (−2.04 to −1.27) mm per month, bootstrap over test years). On seasonal departures the hard form is 5.6 % below the free form (−0.46 (−0.58 to −0.34)). The first figure is close to the 7.4 % found for Jamaica alone in the earlier study with a different network and split (Chapter 7), so that result replicates and extends to two more islands.

Learning adds to the seasonal share when it starts from it. The hard seasonal-departure model is 16 % below the seasonal share (−1.45 (−1.77 to −1.10)). The seed-to-seed spread of the learned models is below 0.1 mm per month for the departure forms, well inside these differences.

The first clause of H1 holds: with an accurate driver, exact aggregation helps.

6.2 Under a controlled bias, every estimator sits just above the floor

The fitted models were then held fixed and the driver multiplied by (1 + α). Proposition 7 says that any estimator that aggregates exactly to the driver has a mean-square error of at least the mean-square driver error. For the seasonal share at α = +0.25 the proposition holds as an identity to rounding error: the mean-square error 1270.2 equals the floor term 1143.8 plus the within-parent variance of the error 126.4 (defect 1.8e−13). In root-mean-square terms the floor is 33.8 mm per month and the part an allocation method can influence is 11.2.

Table 6.2. Error under a multiplicative driver bias, all three islands (mm per month). Fitted models are those of Table 6.1, unchanged.

Figure 6.2. Error against driver bias. The dashed line is the floor of Proposition 7.

A driver bias of 10 % roughly doubles the error of the best estimator, from 7.7 to 15.8 mm per month. At 25 % all three estimators are at most 5.4 % above the floor, and the spread between them, 0.31 mm per month, is a hundredth of the floor. Past a 10 % driver bias the choice of allocation method no longer matters to the total error.

The hard form’s advantage over its free twin shrinks from −0.46 mm per month at zero bias to −0.09 at +50 % and −0.27 at −50 %, and it never changes sign. There is no crossover in this sweep. The reason is in the form of the free twin: it multiplies the driver by a learned factor, so a multiplicative bias passes through it as fully as through the hard form. A model fitted on an exact driver has no reason to have learned a correction, and has none to apply.

6.3 With an independent driver the constraint becomes a liability, if the free model is allowed to learn

The gauge-only analysis differs from the coarsened reference by far more than any allocation error (Table 6.3).

Table 6.3. The gauge analysis against the exactly coarsened reference, test years (mm per month).

Over Jamaica the gauge analysis is 34 % wetter than the satellite and gauge blend. The root-mean-square difference between the two products is 2.4 times the error of allocating uniformly. Which product is nearer the truth cannot be settled here; the point is that two reputable products disagree at the parent scale by more than twice the whole allocation problem.

Table 6.4. Error with the gauge analysis as driver (mm per month). “Perfect framework”: fitted on the exact driver, applied to the gauge driver. “Imperfect framework”: refitted on the gauge driver.

Figure 6.3. With the gauge analysis as driver. Hatched bars are the floor set by the driver’s error.

In the perfect framework every estimator, hard or free, lands between 73 and 76 mm per month, just above the floor of 70.2. The order of Table 6.1 has gone: the hard and free departure forms differ by 0.09 (0.02 to 0.16).

Refitting on the gauge driver separates them. The hard forms cannot move: they must reproduce the driver’s total, and they stay at 72.7 and 75.1, above the floor as Proposition 7 requires. The free forms learn to correct the driver. The free departure model reaches 57.9 mm per month, 18 % below the floor, and is lower than its hard twin by 14.9 (7.3 to 21.4) mm per month. Even the plain free network, which lost to everything with an exact driver, now beats the seasonal share (−10.7 (−16.7 to −4.3)).

The gain is uneven across islands. It is large for Hispaniola (21 % below the floor), modest for Jamaica (6 %) and nil for Puerto Rico (1 %), where the gauge analysis has almost no mean bias to correct.

Two cautions belong with this result. What the free model has learned is a map from one product to another, and it is a correction only if the reference is the better product. And the correction was learned and tested on the same three islands in different years; whether a driver’s bias is stable enough to be learned in one period and applied in another is exactly what breaks when the driver changes version or a gauge closes.

6.4 Transfer to an unseen island

Leaving each island out of training in turn changes the departure models little (Table 6.5). The hard departure model trained on the other two islands is within 0.86 mm per month of the model that saw the island, and beats the island’s own seasonal share by 10 % for Jamaica and 20 % for Puerto Rico. The departure models still read the target island’s own climatological shares, so what transfers is the learned departure from the season and not the season itself. The plain networks transfer worse, by up to 5.8 mm per month.

Table 6.5. Leave-one-island-out error with an exact driver (mm per month). “Trained with” repeats Table 6.1.

6.5 Verdict on H1

H1 said that exact aggregation helps or does no harm with an accurate driver, and that the advantage shrinks and reverses as the driver’s bias grows.

• Accurate driver. Supported on all three islands.
• Shrinks. Supported. Under a controlled bias the advantage falls by between a third and four fifths, and all estimators converge on the floor.
• Reverses. Not in the controlled sweep, where the decision rule looked for it: fixed models show no crossover up to ±50 %. The reversal does occur with an independent driver when the models are refitted to it, and it is large.

By the rule frozen in Chapter 4, H1 is supported in part. The rule was written around the sweep and missed the condition that matters, which is whether the free model has been given the chance to learn the driver’s error. Stated with that condition, the result is sharper than the hypothesis: a hard aggregation constraint is the better choice when the driver is trusted or when the model was fitted on a clean driver, and the worse choice by a wide margin when the model is fitted on the driver it will be used with and that driver is biased.

6.6 What this means for the method

1. Report the driver’s error before the allocation’s. Here it is 2.4 times larger.
2. Start from the seasonal share. A model that does not include it will spend its capacity failing to rediscover it.
3. Conserve to a quantity only as far as that quantity is trusted. This is the allocation-scale version of the weighted ledger of Chapter 3, where the weight on the budget falls as the rainfall product’s error grows.
4. Every figure in this chapter is skill against a satellite and gauge blend. The gauge test of Section 8.3.6 is what would turn it into skill against rainfall.

Chapter 7. Baselines, information sets and foundation-model priors: the component studies

Eight studies were run before the simulation test of Chapter 5. Each asks a narrow question about one component of a Caribbean prediction system, and each was archived and scored before its manuscript was written. This chapter states what each found and what it contributes to the thesis. The figures quoted are those of the component manuscripts (Papers A and C and their supplements), where the full tables, protocols and limitations are given; none was re-run for this chapter. Table 7.1 lists them by the class of evidence they rest on.

Table 7.1. Component studies by data class. An observed product is a satellite–gauge analysis. No study has independent gauge verification beyond one station in the first row.

7.1 Paper A: what a learned rainfall model must beat, and with what information

Paper A consolidates four of the studies around one question: for daily rainfall over Jamaica and neighbouring islands, which baselines and which information sets must any learned model be measured against?

Information age. A small network (1,377 parameters) was given rainfall inputs delayed by zero to 45 days. With fresh inputs it beat climatology. By seven days of delay the advantage was gone, and at 45 days climatology had the lowest error in every row of the comparison. The delays were scenarios set by the analyst and not audited release times of the product, so the result is about stale inputs in general.

Linear comparators. At the Kingston cell, ridge regression and logistic regression matched or beat the network. Recalibration out of period and pooling across cells helped when labels were scarce. One fixed network and one fixed penalty were used, so the finding is that an untuned linear model is a serious opponent, not that linear models win in general.

A frozen foundation model. A time-series foundation model applied with no training gave the lowest next-day mean absolute error on the island series, 13.71 % and 12.36 % below the best local control for its two checkpoints, and never the lowest root-mean-square error. The model is probabilistic and was scored through its median, which favours the first metric and penalises the second by construction; the manuscript says so and names quantile scores as the missing evaluation. It detected 3 of the 15 largest island-mean rain days.

Thermodynamic features. Across six Caribbean cells and 14 test years, adding humidity-based thermodynamic features to a heavy-rain classifier changed the Brier score by −3.5 × 10⁻⁵ with an interval that includes zero. A logistic model on lagged rainfall had the best score on all four metrics, with a Brier skill of 11.7 to 12.1 % over climatology. Predictors and target came from one reanalysis, which the manuscript gives as its principal limitation.

Generative ensembles against the right references. A conditional diffusion model with 1.7 million parameters, driven by satellite imagery, was first reported with a rainfall CRPS 23 % below the best reference. Both references were point forecasts, and the CRPS of a point forecast is its absolute error, which any distribution beats. Rescored against four reference distributions built from the rainfall record alone, the ensemble scored 3.287 mm per day against 3.334 for a climatological distribution, 3.183 for a three-state Markov chain (Gabriel and Neumann 1962) and 3.198 for a lagged ensemble (Hoffman and Kalnay 1983). At the 10 mm threshold its Brier score was 0.1204 against 0.1178 for the climatological frequency. It missed the three wettest test days. I keep the correction in the thesis because the error is a common one and the remedy is cheap: score distributions against distributions (Wilks 2019).

What Paper A fixes for the rest of the thesis. Any rainfall component must be compared with a seasonal climatology, persistence, a ridge or logistic model on lagged rainfall, and reference distributions; its inputs must be those available at issue time; and its verification must name the product it is scored against.

7.2 Paper C: a frozen foundation model against a physical closure for temperature

Paper C asks whether a reduced physical closure adds forecast skill for seven-day 2 m temperature at three Jamaican grid cells when the alternative is a time-series foundation model used with no training (Das et al. 2024).

The study compared 22 models under matched ablations on ERA5-derived targets, with a frozen evidence package from which every table was generated. Two results stand. The frozen foundation model was not beaten: the best trained ensemble was 0.35 % lower in error with an interval that includes zero, which the manuscript reports as no established advantage. And a reduced relaxation–radiation–ocean closure inside a symmetric exchange graph gave its matched graph network a 2.2 % gain in 2026 that did not hold over 2023 to 2026 and reversed in 2024.

Three limits are stated in the manuscript and carried here. The target is a reanalysis product and no station verified it. The evaluation years had been inspected in earlier rounds, so there is no clean hold-out. Every matched effect was under 0.04 °C.

For this thesis the study is a caution about the kind of exchange a simple graph can represent. A symmetric, distance-based exchange between cells is a diffusion. Moisture transport in the trades is advection. A closure that can only diffuse will help where the truth diffuses, and the temperature result is consistent with that reading.

7.3 Two operator studies

Two further studies bear on the central distinction of the thesis, between satisfying a physical statement and being accurate.

In the soil-temperature study a learned flux stencil was placed inside a physics-informed network and fitted to hourly soil temperatures at three USCRN stations (Diamond et al. 2013) over ten station-season blocks. The learned stencil scored 0.591 °C against 0.590 °C for exact finite differences, and a constant-diffusion ablation scored best of all at 0.567 °C. A reconstruction with no network matched the best network in 0.05 s against about 7 s. The operator budget was met and the reconstruction did not improve.

In the closure-tolerance experiment a one-dimensional advection–reaction system was given a wrong wind. The learned source term compensated for it while the ledger stayed exact to 10⁻¹⁶. That is the transport–source counterpart of Proposition 4: with a free source, a wrong transport is invisible to the budget. It is the reason the unresolved tendencies in Chapter 3 are prescribed or bounded and never free.

7.4 What the component studies establish together

1. Simple, local, climatological baselines are strong at these sample sizes and must be in every comparison.
2. A frozen model trained elsewhere can be the strongest reference available, at no training cost.
3. A constraint can be met exactly while accuracy does not move.
4. The reference products carry errors as large as the differences between methods. At the single gauge available to the rainfall-allocation study, the reference product’s own error exceeded every model’s.

None of the eight is the atmospheric reconstruction. Together they are the reason the hypotheses of Chapter 4 were written with decision rules that allow them to fail.

Chapter 8. Application to Jamaica: protocol and calculations awaiting local data

Stage 2 applies the method to observed events over Jamaica with station data. This chapter fixes what will be computed, on what data, and what would count as success. Every table that needs local data is laid out here with its cells marked [awaiting data], so that the calculations can be run and the chapter completed without changing its design. Nothing in this chapter is a result.

8.1 What changes from Stage 1

The second row is the important one. With ERA5 as the background the room for improvement is smaller than in Stage 1, where the background was deliberately rough, and the result of Chapter 5 that the conventional smoother does as well as the learned variants becomes the prior expectation.

8.2 Data required

Table 8.1. Data needed for Stage 2 and their status.

8.3 Calculations to be run

8.3.1 Observing-network table. Replace Table 2.1 with counts from the national archives.

8.3.2 Closure of ERA5 over the domain. Repeat the closure-floor calculation of Section 5.1 with ERA5 as the reference and its own forcing, for the event regimes of Table 4.0. This is public data and is the first Stage 2 calculation to run. Expected outcome from Chapter 5: usable information only at two degrees and three hours or coarser.

8.3.3 Error statistics from stations. Station-minus-ERA5 statistics over a training period that excludes the events: bias-like and structured standard deviations, length and time scales; representation error by station class.

8.3.4 Event catalogue. The rule-based catalogue of Chapter 4, restricted to events with at least the minimum number of reporting stations set in 8.4.

8.3.5 Reconstruction and verification. For each event: B0, B1, B3, C1 and C3 at the full network and with stations withheld.

8.3.6 Allocation at gauges (H1). Repeat Chapter 6 for Jamaica with verification at gauges that do not enter CHIRPS, and with ERA5 and a seasonal forecast as real drivers.

8.4 Decision rules for Stage 2

These are fixed now, before the data arrive.

1. Minimum network. An event enters the catalogue only if at least ten surface stations report through the window and at least one sounding falls inside it. The number is provisional and will be revised once only, when the station inventory of 8.3.1 is known and before any reconstruction is run.
2. Withholding. One station in four is withheld by a seeded draw stratified by parish, and never read until scoring. Soundings are withheld by ascent.
3. Primary outcome. CRPS of surface specific humidity at withheld stations, relative to the background.
4. Success for the ledger. B3 or C3 improves the primary outcome over its no-ledger control with an event-bootstrap interval excluding zero. From Chapter 5 the expected effect on a point humidity observation is small, because the ledger acts on block-scale divergence, and a null result here would not contradict Stage 1.
5. Success for coupling. As H2: C3 against C1 at the sparsest network.
6. What will not be claimed. That an unobserved transport field is validated because it agrees with the background. That skill against CHIRPS is skill against rainfall.

8.5 Risks

• The station archive may hold too few stations with complete metadata for a withheld set of useful size. The 1992 archive fire is the known cause of short series.
• Surface humidity constrains the lowest layer and the transport sits higher, in the jet. Without GNSS or more soundings the column form has little to assimilate over Jamaica, and the pressure-level class of Section 3.12 becomes necessary and not optional.
• If most gauges enter CHIRPS, the independent set for H1 may be a handful of gauges, and the allocation result will remain a result against a product.

Chapter 9. Discussion

9.1 What the evidence supports

The thesis set out to test whether a shared-face conservation ledger, carried by small neural networks on space-time bricks, improves the reconstruction of atmospheric moisture and rainfall over the Caribbean. Three hypotheses were written down with decision rules before the test data were opened. Table 9.1 gives the outcome.

Table 9.1. Hypotheses, decision rules and outcomes.

The principal hypothesis failed. The method as named, flux-coupled neural bricks, is not better than its alternatives at the things it was built to do. I report that as the main result of the thesis and not as a footnote to it.

What survived is a narrower claim with better support than the original would have had:

A discrete, weighted moisture ledger on coarse control volumes improves the block-scale divergence of a reconstructed transport field, from rainfall and column vapour alone, and how the field is represented does not matter.

Its support is as follows. The effect is large beside its uncertainty (−0.324 (−0.438 to −0.227) for the smoother and −0.236 (−0.310 to −0.171) for the shared-flux blocks, against their own no-ledger controls). It is present at every network density and without any sounding. It appears in a grid method and in every neural variant. It survives a 25 % bias in the rainfall product nearly intact and a 50 % under-estimate at about half strength. And it is bounded in a way that was measured in advance: the truth’s own budget residual is 0.73 of the background’s at the control-volume size used.

9.2 Five things I did not expect

Assimilating sparse transport observations without a budget makes divergence worse. Interpolation of two soundings raised the block-scale divergence error to 1.11 of the background’s. A local correction to a vector field is a divergence dipole. Any regional analysis that nudges transport towards isolated soundings should check its divergence.

Exact cancellation is bookkeeping. The shared-flux variant cancels interior transfers to rounding error and the state-matched variant leaves a quarter of them uncancelled, and their accuracy is the same. Proposition 2 is true and, for accuracy, beside the point: a verification that differences the assembled field does not care whether two blocks agreed on a number.

Sharing a flux does not make a field. The shared-flux blocks have the largest jumps at their faces and a grid-scale divergence error a quarter above the background’s. Conservation across a face and continuity across a face are different properties, and the method needs both. The exploratory variant that has both was the best arm in the test on every divergence measure, by margins too small to claim.

Where the stations are matters more than how many there are. Twelve sites gave worse column vapour than six for every neural variant, because the stations sit over land in the north where the background is nearly unbiased and over the open water to the south its bias is −3.0 kg m⁻². A network designed for airports samples the atmosphere’s errors badly.

A constraint should be as soft as its data are uncertain. In Chapter 6 an exact aggregation constraint passed a driver’s bias through in full and lost to a free model by 15 mm per month. In Chapter 5 a weighted budget constraint lost half its benefit under a 50 % rainfall bias and kept the rest. The two results are the same principle seen from opposite sides, and the principle is the practical content of the thesis: conserve to what you trust, with a weight that says how much.

9.3 What this means for the method

D-LEGOS should not be presented as a better solver. On this evidence a variational smoother on a grid with the same ledger term does as well on divergence, better on column vapour, and is simpler to reason about. What the brick construction offers is speed, 6 times that of a single network of the same size, and a formulation in which the ledger, its weights and its limits are explicit. The nine propositions of Chapter 3 held in every numerical check, and three of them (the blindness of a budget to circulation, the bound under driver bias, and the compatibility of an exact ledger with wrong transfers) predicted the main empirical results before they were run.

The recommendation for anyone reconstructing regional moisture transport is therefore plain. Put a weighted block budget in the objective, at a control-volume size where the reference product has been shown to close; use whatever representation is convenient; do not expect the transport itself to improve without observations of the column over the sea; and verify with a discretisation that the fit never saw.

For the component studies of Chapter 7 the lesson is the same one in other clothes. A seasonal climatology, a linear model and a frozen foundation model were not beaten at these sample sizes, and each physical constraint was satisfied without accuracy moving. In Chapter 6 a lookup table removed 68 % of the allocation error before any network was trained.

9.4 Assumptions

The results rest on the following assumptions. Each is stated with what would change if it failed.

Physics.

1. Column budget. Condensate storage and transport are neglected, vapour flux through the model top is neglected, and the column integral runs from the product’s surface pressure (Section 3.3). If condensate transport matters, as it does in the outflow of a cyclone, rain is booked in the wrong block or hour and the ledger residual is not zero for the true fields. The closure floor of Section 5.1 measures the combined size of these terms for ERA5 and is the reason the ledger is used at 2° and 3 h.
2. Evaporation is known. Evaporation is taken from the background. Over the sea in the trades its error is small beside that of rainfall; over land after rain it is not.
3. Unresolved tendencies are bounded. Nothing in the objective is free to absorb a transport error. This is a design choice forced by the wrong-wind experiment of Chapter 7.

Statistics.

4. Background errors are Gaussian, stationary and separable in space and time, with two components (a domain-wide offset and a structured part) whose scales were estimated on development events. A cyclone violates stationarity by construction.
5. Observation errors are unbiased and uncorrelated between sites and times. Real surface stations have correlated representation errors.
6. The rainfall product’s error is multiplicative and lognormal with a fixed correlation length of three cells. Real satellite products miss events outright and place rain in the wrong cell, which a multiplicative error cannot represent.

Experimental.

7. ERA5 is the truth. It is a model product with its own budget residual. The test asks whether a method can recover ERA5 from a rougher reanalysis and sparse samples of ERA5, which is a necessary condition for recovering the atmosphere and not a sufficient one.
8. Observation operators are exact. Synthetic observations are sampled from the nature run by the same bilinear interpolation the reconstruction uses, so there is no operator error. The background comes from a different model, which avoids the identical-twin problem for the dynamics, but not for the observations.
9. The fine rainfall reference is CHIRPS. Chapter 6 measures skill at reproducing a product.

9.5 Limitations

Of the formulation.

• A ledger constrains divergence at the block scale. It does not see the rotational part of the transport (Proposition 4), which is most of the transport in the trades.
• The shared-flux variant does not make the state continuous across block faces. Section 5.3 shows the price at the grid scale.
• The pressure-level class of Section 3.12, in which wind, humidity and mass are reconstructed on levels with a three-dimensional ledger, is formulated and was not run. Every Stage 1 statement is about the column form.

Of the simulation test.

• Fifteen test events of 48 hours over one domain. Regime-level statements rest on three events each.
• One nested thinning of the surface network. The effect of network density is confounded with which sites happened to be kept, and Section 5.2 shows that this matters.
• Penalty weights were tuned over a small grid on five development events and two densities. The 50 % and 10 % networks were never seen in development.
• The optimiser budget was fixed at 300 iterations. An audit at 900 iterations is reported, on one variant only.
• Ensembles of eight members. Coverage estimates from eight members are coarse.
• No convection-permitting nature run. Structure finer than 0.25° is absent from the truth, so nothing here bears on kilometre-scale rainfall.

Of the allocation experiment.

• The reference is a satellite and gauge blend. Where no gauge is near, its fine structure is an infrared proxy shaped by a climatology, and a seasonal share will reproduce that climatology by construction.
• The gauge analysis used as an independent driver is not fully independent of the reference: both draw on the same public gauge reports where they exist. Their errors differ in source over ungauged terrain and may be correlated near gauges.
• Six test years.
• Parents are registered half a fine cell west of the gauge-analysis cell edges (Appendix A).
• Hispaniola supplies three quarters of the parents, so pooled numbers are mostly Hispaniola.

Of the component studies. Gridded targets throughout, one station for verification in total, evaluation periods that had been inspected before, and effects smaller than plausible product error (Chapter 7).

Of the thesis as a whole. No result in it has been verified against an independent Jamaican gauge or sounding. Chapter 8 exists because of that.

9.6 Threats to validity

Internal. The plan was frozen before the test events were run, the test was run once, and the one code change after the freeze is listed in Appendix A with its timing. Development and test events are from different years. The remaining threat is tuning asymmetry: the block-decomposed variants were tuned over three values of one weight and the variational smoother over its own small grid, and a method tuned harder could have gained a few percent.

Construct. The primary outcomes are errors in transport and in block-scale divergence, relative to the background. Block-scale divergence was chosen because it is what the budget constrains and what links transport to rainfall, but it is one step removed from rainfall itself. A method that improves it has not been shown to improve a rainfall estimate.

Statistical. Fifteen events give wide intervals. Two primary outcomes were tested at four densities with a decision rule fixed in advance and no correction for multiplicity; a reader who prefers a correction should treat intervals that barely exclude zero as inconclusive. The event bootstrap treats events as exchangeable, which ignores that three test events fall in one November.

External. Real observations have biases, gaps and correlated errors that the simulation lacks, and the real atmosphere has scales the nature run lacks. Both differences make the real problem harder. The simulation result is therefore an upper bound on what to expect from the same method on real data, and where the simulation shows no gain, none should be expected.

9.7 Gaps and further work

1. Stage 2. The protocol of Chapter 8, starting with the closure of ERA5 over the domain, which needs no local data.
2. Network geometry. Repeat the density experiment over many random thinnings to separate density from geometry, and replace the low-order offset model with one that cannot extrapolate into unobserved water.
3. Observations of the column. GNSS column vapour and satellite microwave retrievals observe W over the sea, where the surface network is empty. Their assimilation is the most direct route to a better transport.
4. The pressure-level class. Needed wherever surface humidity is the main observation, because the transport sits in the jet above the layer that stations sample.
5. Rainfall error models that include missed and misplaced events, and a ledger weight that follows a time-varying estimate of the product’s error.
6. Independent gauges for the allocation problem, and a real coarse driver (a seasonal forecast) in place of a coarsened reference.
7. Continuity and conservation together. The variant with shared flux and state matching (C4) was included as an exploratory arm. Its behaviour in Chapter 5 makes it the natural default for the next round.

Chapter 10. Conclusions

10.1 Answers to the research question

The question was whether conservation, imposed through shared faces between space-time control volumes, improves the reconstruction of atmospheric moisture and the allocation of rainfall over Caribbean islands when the coarse information driving the reconstruction is itself in error.

1. A weighted moisture ledger improves block-scale divergence, and that is all it improves. In a simulation test over the Caribbean with ERA5 as truth and an older reanalysis as the first guess, every method that carried the ledger lowered the error in 2° divergence, to 0.81 to 0.83 of the background’s for all but uncoupled blocks (0.91), and every method that did not carry it ended above the background. Transport improved by at most 2.5 % and its rotational part did not improve.

2. The brick construction is not what does the work. Shared-flux neural blocks, state-matched blocks, a single network and a variational smoother on a grid gave the same primary outcomes. The principal hypothesis, that shared-flux coupling would beat state matching under sparse observations, was not supported.

3. Exact conservation helps only as far as the conserved quantity is right. For monthly rainfall on three islands, forcing the fine field to aggregate to the coarse driver lowered error by about 6 % with an exact driver, made no difference for a model trained on a clean driver and applied to a biased one, and cost 15 mm per month against a free model when both were fitted to a biased independent driver. The error of that driver was 2.4 times the whole allocation problem.

4. A weighted constraint degrades gracefully where an exact one does not. The same ledger kept about half its benefit when the rainfall product under-reported by half.

5. Simple references set the bar. A seasonal lookup removed 68 % of the allocation error; in the component studies a climatology, a linear model and a frozen foundation model were not beaten.

10.2 Contributions

• A formulation of flux-coupled space-time control volumes for moisture reconstruction, with nine propositions stating what a discrete ledger guarantees and what it cannot, each verified numerically (Chapter 3, Appendix B).
• A measurement of the scale at which the column moisture budget of ERA5 carries information over the Caribbean (Section 5.1).
• A simulation test under a frozen plan that separates the effect of the ledger from the effect of the representation, with a negative result on the principal hypothesis (Chapter 5).
• A bound on the error of exactly aggregating downscaling estimators under driver error, and a three-island experiment that locates when the constraint helps and when it harms (Chapters 3 and 6).
• A set of baseline and information-set results for Caribbean rainfall and temperature prediction (Chapter 7).
• An open, documented code base in which every reported number is regenerated from stored results and checked mechanically (Appendices A and D).

10.3 What remains

No result here has been checked against an independent Jamaican gauge or sounding. The simulation used the column form only. The protocol of Chapter 8 states what is to be computed when local data are in hand, on which outcomes, and what would count as success. From Chapter 5 the prior expectation for that work is modest and specific: a gain in block-scale divergence where rain falls, none in transport without observations of the column over the sea, and no advantage for neural blocks over a conventional smoother with the same ledger.

The useful next steps are the ones the negative results point to. Observe the column over water, with GNSS and microwave retrievals. Test network geometry across many random thinnings. Carry forward the variant with both a shared flux and a continuous state. And replace the fixed ledger weight with one that follows the rainfall product’s estimated error, so that the method conserves to what it has reason to trust.

Appendix A. Code, data and reproducibility

A.1 Layout of the repository

A.2 Data and access routes

Every downloaded array is stored with its SHA-256 hash. The frozen plan records the hashes of the calibration file, the event catalogue and the source files that produce Stage 1 results.

A.3 Order of operations and what was frozen

1. Fetch and hash the data (scripts 01 and 03).
2. Build the event catalogue by rule (script 02).
3. Estimate error statistics and the closure floor on development events only (script 04).
4. Tune penalty weights on development events only (script 05).
5. Freeze the plan: settings, hypotheses, decision rules, smallest effects of interest, file hashes (script 06, frozen at 2026-11-11T09:02:05Z UTC).
6. Run the test events once (script 07).
7. Analyse, draw figures, render the manuscripts from the result files and run the checks (scripts 09, 12, 13 and 14).

The allocation experiment (scripts 06b and 08) has no tuning stage; its training, validation and test years were fixed in the code before any model was fitted.

A.4 Changes made after the plan was frozen

No Stage 1 source file was changed after the freeze.

A.5 Computing

All experiments ran on two CPU cores with no accelerator. One block-decomposed reconstruction of a 48-hour event takes about ten seconds, the variational smoother about twenty and the single global network about a minute. The Stage 1 test (15 events, four network densities, eleven methods, plus stress tests and ensembles) takes about four core-hours. The allocation experiment takes under one core-hour.

Appendix B. Numerical checks (Stage 0)

These tests run before any experiment and must pass on the machine that runs the experiments (python -m pytest tests/).

Verification in Chapter 5 uses a second, independent discretisation (centred differences on the grid, not the finite-volume faces used in training) for divergence and curl, so that no diagnostic repeats the objective.

Appendix C. Event catalogue

Table 4.0. Event catalogue. Score: best-track wind in knots (TC), domain-mean rain in mm per day (EARLY, LATE, DRY), 925 hPa zonal wind in m s⁻¹ in the low-level-jet box (MSD).

Appendix D. Claim-check record

Numbers in this thesis are not typed. Each is a placeholder in the source that the build step fills from a stored result file, and each substitution is written to a claim specification. The BootLoops Emitall engine then re-reads the result files and compares every printed value at its printed precision. Separately, the headline Stage 1 ratios are recomputed from the raw result rows, and the two allocation estimators that involve no learning are recomputed from the raw island files, with code that shares nothing with the analysis scripts. A negative control plants three errors in a copy of one specification and requires the engine to flag exactly those three. The record of the final run is in bootloops/CHECKS.md, distributed with the thesis, together with the reference-check record (bootloops/REFERENCE-CHECKS.md: 79 of 89 references resolved against Crossref, DataCite or arXiv by identifier and title; the other ten are books, reports and web pages with no identifier, checked by hand).

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