The eight papers in plain language

What is D-LEGOS?

D-LEGOS is a proposed way to reconstruct atmospheric conditions from incomplete observations. Local neural fields represent different regions. Equations and observations constrain those fields, and neighbouring regions share one record of what crosses their boundary.

Paper P7 explains the programme. It contains completed mathematical derivations and a small constructed code test. The atmospheric reconstruction is still proposed. Producing a finer grid would not, on its own, establish accurate sub-kilometre information.

The other seven papers stand on their own. They test specific questions about rainfall, temperature, information and numerical methods. The explanations below describe those questions without turning them into completed atmospheric applications.

Source: P7 manuscript. References.

P1: Rainfall totals and local allocation

Source: P1 manuscript. References.

P2: Additional inputs for temperature forecasts

Source: P2 manuscript. References.

P3: Delayed rainfall records and limited labels

Source: P3 manuscript. References.

P4: Daily rainfall error and heavy events

Source: P4 manuscript. References.

P5: Rainfall transport, growth and decay

Source: P5 manuscript. References.

P6: Soil heat flux and reconstruction error

Source: P6 manuscript. References.

P7: Shared fluxes and atmospheric reconstruction

Source: P7 manuscript. References.

P8: Thermodynamic features and transfer between cells

Source: P8 manuscript. References.

Glossary

These definitions describe the terms used across the research pages. Follow each manuscript for its exact assumptions and equations.

Ablation

A comparison where one component changes while the rest of the test stays as similar as possible. Removing the physics penalty, for example, helps isolate what it contributed.

Adaptive pathway

A sequence of decisions that can change as conditions or evidence change. For example, a plan can specify the conditions that would trigger a drainage upgrade.

Advection

Movement of a quantity by a flowing medium, such as wind carrying water vapour or moving an existing rain pattern.

Area-weighted mean

The average for a large cell, calculated so each smaller area contributes in proportion to its size. Many patterns inside the cell can share the same average.

Asynchronous updates

Updates that take place at different times. A local model may then act on older information from a neighbour.

Atmospheric forcing

Conditions supplied to drive a model, such as rainfall or wind that drives a surface response. An imposed storm is specified as an input to the scenario.

Atmospheric reconstruction

An estimate of a hidden or past state using available measurements. Retrospective reconstruction may use measurements taken after the time being estimated, so it differs from a forecast issued beforehand.

Atmospheric state

The description of the air at a given place and time, such as its temperature, pressure, humidity and wind. “Moist” means that the model accounts for water in the atmosphere.

Atmospheric world model

A representation of an atmospheric state and its evolution. P7 proposes connected local neural fields for retrospective reconstruction. The current formulation has not been validated as a complete atmospheric world model.

Baseline

A reference method a new model has to compete with under the same test conditions. A well-chosen simple baseline shows whether extra complexity earns its place.

Budget residual

The difference between how much a stored quantity changed and how much the recorded transfers and sources say it should have changed. It shows where the accounting fails to close.

CHIRPS

Climate Hazards Group Infrared Precipitation with Stations. It combines satellite information and station data into gridded rainfall estimates. Agreement with CHIRPS measures agreement with that product.Funk et al. on CHIRPS

Climate adaptation

Changing practices, infrastructure or plans in response to current or expected climate conditions. An adaptation option is one possible change to assess.

Climate hazard

A climate-related event or condition that could cause harm, such as intense rainfall or extreme heat. What happens also depends on who or what is exposed.

Climate projection

An estimate of how climate statistics could change under stated assumptions about future conditions, such as greenhouse gas emissions.

Climatology

Typical conditions calculated from a historical record. A seasonal climatology can give a simple forecast reference.

Conductance

In the reduced heat test, the coefficient that links the temperature difference between two cells to the rate of heat exchange. A fitted value is chosen using training cases.

Conservation

A balance between stored quantities, boundary transfers and sources. P1 conserves a supplied rainfall aggregate; P5 and P7 define numerical transfer ledgers. These different constraints do not by themselves prove physical accuracy.

Constructed test

A constructed system whose data and governing rules are specified by the experiment. P5 uses a one-dimensional synthetic transport-and-growth test; P7 uses a manufactured heat-exchange chain.

Continuous ranked probability score

A score for a whole forecast distribution. It compares the predicted probabilities with the observed value. Lower values indicate better agreement on the cases scored.Gneiting and Raftery on probabilistic scoring

Control volume

A defined piece of space with a boundary. You can account for how much water or heat is inside, how much crosses its boundary and how much comes from a source within it.

Critical success index

A score for events such as heavy rain. It divides correct event predictions by correct predictions plus missed events and false alarms.

Data assimilation

Combining measurements with a model estimate to infer the state of the atmosphere. Four-dimensional variational assimilation fits the state across a time window. Its weak-constraint form allows for model error.Trémolet on weak-constraint 4D-Var

Decision regret

The difference, within one scenario, between the result of the chosen action and what the best action in that scenario would have achieved.

Delayed, unaided transfer

An assessment given after a delay that asks learners to apply the same rule to an unfamiliar problem without help.

Diffusion

A process that tends to smooth differences, such as heat spreading from a warmer region towards a cooler one. The graph test applies this idea to connected cells.

D-LEGOS

Adrian Dunkley’s formulation for reconstructing past atmospheric states with connected local neural fields. Each shared face has one signed transfer record. Paper P7 contains mathematical and constructed-code results; the atmospheric reconstruction remains proposed.Thesis formulation

Downscaling

A way to use a broad weather map to estimate a more detailed map. The model can use the shape of the land and local readings. Local measurements show if the extra detail is right.

Dynamical downscaling

Running a regional physics model on a finer grid, using conditions from a larger-scale model to drive it. It is a different route to local detail from learning a statistical mapping.Taylor et al. on PRECIS Caribbean

Emulator

A simpler model that approximates a more expensive simulator or one of its processes. It needs checking against its reference simulator and, for claims about real weather, against observations.Beucler et al. on neural emulators

ERA5 reanalysis

ERA5 is a historical atmospheric dataset from the European Centre for Medium-Range Weather Forecasts. It uses a consistent weather model and past observations to estimate conditions across a grid.Hersbach et al. on ERA5

Exposure and vulnerability

Exposure asks who or what lies in a hazard’s path. Vulnerability asks how susceptible those people or assets are to harm.

Finite-volume method

A way to solve physical equations by dividing space into cells and accounting for exchanges through their boundaries. The same internal transfer appears with opposite signs in the two cells.

Flux

The amount of heat, water or another quantity that passes through a boundary in a given time. A sign shows its direction.

Forecast calibration

Whether predicted probabilities agree with frequencies over many comparable cases. If a model often assigns a 30% chance of rain, about 30% of those cases should have rain in a calibrated forecast.

Forecasting

An estimate of what may happen after a set time. The forecast uses information available when it starts.

Foundation prior

A starting estimate or starting information from a model trained elsewhere. A frozen model keeps its learned settings fixed while a local method uses or adjusts its output.

Fractions skill score

A comparison of how much of each nearby area was forecast to exceed a rainfall threshold and how much actually did. Changing the neighbourhood size shows the scale at which the forecast agrees.Roberts and Lean on the fractions skill score

Graph coupling

Connecting local models as a network. The nodes are the cells and the edges are the links between neighbours. A link can carry information or represent a defined physical exchange.

Hard constraint

A rule that the calculation is built to satisfy. For example, it can rescale a fine rainfall map so that its area-weighted average matches the supplied coarse mean.

Heat capacity

How much heat is needed to raise the temperature of a cell or material by a given amount. Two cells can exchange the same amount of heat yet change temperature by different amounts.

Holdout

Data or whole events reserved for testing. A holdout is kept away from training and tuning so it can measure how well the method works on cases it has not seen.

Hydrostatic balance

An approximation in which the vertical pressure difference supports the weight of the air above. The study must choose where this approximation is appropriate.

Identifiability

Whether the available information can distinguish one explanation from another. A coarse rain average cannot distinguish fine patterns that all average to the same value. Their invisible differences form a null space.

IMERG

Integrated Multi-satellitE Retrievals for GPM. It combines satellite measurements to estimate precipitation and, in some versions, incorporates gauge information.NASA IMERG documentation

Incidence matrix

A table that records which cells meet each edge and which end has a positive or negative sign. It makes it possible to apply one flux with opposite signs to the two cells.

Instructional scaffold

A temporary hint or guide that helps a learner practise. An unaided assessment removes that help to see what the learner can do independently.

Intervention effect

The change caused by taking an action compared with what would have happened without it. Estimating this requires evidence about both cases.

Issue time

Issue time is when a forecast is made. Lead time is how far ahead it looks. A forecast issued at noon for 3 p.m. has a three-hour lead.

JEPA

Joint-embedding predictive architecture. It learns to predict a compact numerical description of a hidden part from the visible context. A separate step tests whether that description helps recover real quantities such as wind or humidity.Assran et al. on I-JEPA

Latent representation

A compact set of numbers a model learns to describe its inputs. A latent target describes the hidden part the model is trying to predict.

Local field

A description of a quantity such as temperature or wind across one model block. A learned local field estimates how that quantity varies across the block and through time.

Masked reconstruction

A training task that hides part of the input and asks the model to recover it using the part it can see.

Mean absolute error

The average size of prediction errors, with overestimates and underestimates both counted as positive magnitudes. It uses the same units as the measured quantity.

Moisture transport

The movement of water through the atmosphere. Wind carries water vapour, and a moisture budget also accounts for changes such as evaporation and precipitation.

Multiscale

Using more than one scale, such as broad regional conditions together with local detail around a coast or mountain.

NASA POWER

NASA’s Prediction Of Worldwide Energy Resources service. It provides historical weather and solar datasets. A value requested at a coordinate can represent a grid cell rather than a station.NASA POWER documentation

Neural network

A model that learns from examples. It uses layers of calculations to turn inputs such as place and time into estimates such as wind speed.

Nowcasting

Predicting what will happen over the next few minutes to hours using recent information, such as the motion and growth of an observed rain system.Zhang et al. on NowcastNet

Numerical agent

A local calculation that holds a cell’s state and exchanges defined values with its neighbours.

Observation operator

A rule that makes a model estimate comparable with a measurement. It can sample the model at a gauge or average its small cells over the area a satellite pixel represents.

Observation support

The area and time interval a measurement represents. A point gauge, a satellite pixel and a monthly grid average describe different portions of the atmosphere.

Observing-system simulation experiment

A controlled test using a simulated atmosphere with a known state. Researchers give the method only simulated readings and check whether it recovers the hidden parts before testing real events.

OISST

Optimum Interpolation Sea Surface Temperature. This NOAA product combines information from satellites and other measurements to estimate the temperature of the ocean surface on a grid.NOAA OISST documentation

Paired uncertainty interval

A range for a difference between methods evaluated on the same cases. Resampling whole events or time blocks preserves some of the dependence between nearby readings.

Partial differential equation

An equation that relates how a quantity changes across directions or through time. Weather equations use this kind of relationship to describe motion and changes in temperature, pressure or moisture.

Persistence

A forecast that carries forward the latest known value or pattern. A rainfall persistence forecast keeps the latest rain pattern in place.

Physics penalty

An added cost during training when a model output breaks a physical rule. It encourages the model to follow the rule. A remaining error can still occur.

Physics-informed AI

An AI model can learn from data and use rules from physics. One rule can check that the energy that leaves one block enters the next.Beucler et al. on exact and soft constraints

Physics-informed neural network

A neural network whose training or design uses the equations of the system it is modelling. For example, its training can penalise a predicted flow that leaves a water budget out of balance.Raissi et al. on physics-informed neural networks

PISR

Physics-informed super-resolution. DLEGOS-PISR is the proposed test of whether local, connected physics-informed models can recover finer atmospheric fields from coarse Caribbean inputs.Caribbean benchmark design

Pressure level

A layer where the air pressure has a specified value. Since the height of that pressure can change across the region, it differs from a layer at a fixed height.

Rain gauge

An instrument that measures rainfall at a location. An independent gauge check uses readings that were kept separate from the model inputs and the product being evaluated.

Regional background

A model’s broad starting estimate of the atmosphere before local readings refine it. A systematic error in that estimate can pass through to the local reconstruction.

Root mean square error

An error score that squares individual errors before averaging and then takes the square root. It puts more weight on large misses and returns to the original units.

Scenario

A specified set of conditions used to test what might happen. A paired comparison holds the conditions fixed and changes one decision or intervention.

Seasonal allocation

A baseline that shares a coarse monthly rainfall estimate across smaller cells using the historical pattern for that month or season.

Sensible heat

Heat associated with a temperature change, as opposed to the energy used in a change of phase such as evaporation.

Sensitivity analysis

Checking how a conclusion changes when an input or assumption changes. A decision that changes after a small cost adjustment needs special scrutiny.

Shared face

The boundary where two neighbouring model cells touch. A transfer out of one cell across this face is a transfer into the other.

Shared flux ledger

A single record of a physical exchange, including which cells share it, the direction, units and time interval. Both cells use the same entry in their budgets.

Source traceability

The ability to follow a conclusion back to the measurements, models, assumptions and revisions behind it.

Space–time

A way of describing where something happens and how it changes during a given interval. A DLEGOS block tracks a local region from the start to the end of an interval.

Sparse observations

Measurements that cover only some of the places, times or heights of interest. A rain gauge can report its location while leaving the hills between gauges unsampled.

Statistical downscaling

Using relationships learned from data to estimate local conditions from a broader map. A finer rainfall estimate needs local checks even when it matches the broad-area average.Harder et al. on climate downscaling

Super-resolution

Estimating a finer grid from coarser inputs. In atmospheric science, the sharper estimate can be plausible while still missing local details, so it needs independent checking.

Surface storage

Water held on the surface during an event. A model updates that amount as rain arrives and water drains, runs off or infiltrates.

Tesseract

A cube has three dimensions. A tesseract has four. In DLEGOS, the fourth dimension is time. The shape describes a block of air over a period.

Time step

The amount of simulated time advanced in one update. In a numerical model, a step that is too large can produce oscillations or growing errors.

TimesFM

A pretrained time-series forecasting family. P2 uses frozen TimesFM models as temperature comparators; it does not retrain their checkpoints.Das et al. on the TimesFM model family

WeatherNext

A Google DeepMind family of learned weather forecast models. The temperature paper treats archived WeatherNext output as a distinct external comparison.Google DeepMind WeatherNext

Timestral

The pretrained time-series model family tested in P4. Two frozen checkpoints use rainfall history without local fine-tuning; their performance is compared with four local controls.

Brier score

The mean squared difference between a predicted event probability and the observed outcome, coded zero or one. Lower scores are better for the same event and evaluation data.

USCRN

The NOAA U.S. Climate Reference Network. P6 uses its hourly observations from three US stations to test soil-temperature reconstruction.

Physical closure

A specified relation for a process that the resolved variables do not describe completely. P2 uses a reduced temperature relation; P5 tests the limits of a source relation for rain growth and decay.

Tolerance band

A range of departure from a nominal relation that attracts no penalty. In P5, the proposed tolerance is set from development data before test evaluation.

Operator budget

A limit on the error in a local calculation, such as a face flux. Meeting it does not guarantee that the final reconstructed field has low error.

Ridge regression

A linear model whose fitted coefficients are penalised to limit their size. It is a local baseline in the rainfall studies.

Logistic regression

A statistical model that predicts a probability. P3 and P8 compare neural or boosted predictions with logistic baselines.

Observation latency

The time between the event represented by an observation and the time that observation becomes available. The fixed delays in these studies are test scenarios, not reconstructed publication histories.

Oracle-assisted gate

A safeguard that can consult information unavailable to an ordinary learned approximation. P6 checks the learned flux against the known exact map and its derivative.

Block bootstrap

An uncertainty calculation that resamples groups of consecutive times or other clusters. The block definition and the uncertainty excluded from the calculation must be checked in each study.