Tier III — Eulerian finite-volume transport

Forward model: full PDE solved on a grid. Uses finitevolX for spatial discretisation and spectraldiffx for spectral / periodic-domain operators.

This is the gold-standard physics tier: full mass conservation, arbitrary wind fields, support for chemistry and deposition. Also the most expensive — emulators (Step 3) are essential, not optional.

(1) Simple model¶

Conservation equation¶

Operational implementations carry mixing ratio $c$ (kg/kg), not mass density, because that’s what the satellite observes and what WRF couples to. The conservation form is then:

with $\rho(\mathbf{x},t)$ the dry-air density from the met field. When ρ is approximately constant over the integration window (typical at fixed altitude in a regional basin) this reduces to the textbook $\partial_t c + \nabla\cdot(\mathbf{u}\, c) = \nabla\cdot(\mathbf{K}\nabla c) + S - \lambda c$. Commit to the full ρ-weighted form for any work crossing significant pressure-altitude variation.

• $\mathbf{u}(\mathbf{x},t)$ — wind from WRF / ERA5 / HRRR.
• $\mathbf{K}(\mathbf{x},t)$ — eddy diffusivity. Commit to K-theory from MO similarity in the PBL ($K_z \approx \kappa\, u_*\, z (1 - z/L)$ with stability correction; Monin & Obukhov, 1954Stull, 1988) plus Smagorinsky in the free troposphere. Source: MO prereqs + WRF TKE.
• $S(\mathbf{x},t)$ — source field, the unknown we invert for. Per-cell, time-resolved.
• λ — first-order chemical loss. Negligible for CH₄ at <1-day scales; included for portability to CO/HCHO.

Initial and boundary conditions¶

Both are load-bearing for inversion accuracy and currently underspecified in operational code:

Spatial / temporal discretisation¶

Cell-centred FV with flux-limited advection via finitevolX; explicit RK or IMEX for stiff diffusion; spectraldiffx for periodic / global domains.

Build order within Tier III¶

1. 1D column model. Validates numerics — diffusion against analytical Gaussian, advection against cosine pulse / Burgers’.
2. 2D horizontal slab. What most satellite work needs (vertical column already integrated by AK).
3. Full 3D with vertical layers. Needed when injection altitude matters or when assimilating multi-layer in-situ data alongside columns.

(2) Model-based inference¶

Forward observation operator¶

Same column + AK pipeline as Tiers I and II:

$\mathbf{H}_t \coloneqq \mathbf{A}_t \cdot \mathrm{col}_z(\cdot)$ is the observation operator at time $t$. The 4D-Var cost below uses $\mathbf{H}_t$; the column + AK implementation is shared with Tier I and the prereqs.

4D-Var cost — three terms¶

Three terms — source background, IC background, and time-summed observation mismatch. The IC term drops out only if you commit to a long warm-up that pins $c(0)$. Treating observations as a single instantaneous mismatch (no time index) is incompatible with multi-hour assimilation windows.

Likelihood model¶

• $\mathbf{R}_{\text{retr},t}$ — heteroscedastic per-pixel from the L2 retrieval-error map at time $t$.
• $\mathbf{R}_{\text{repr},t}$ — representation error (model-vs-observation footprint mismatch). Diagonal addition; rises with terrain complexity and at coarse-instrument boundaries. Don’t omit — naive $\mathbf{R} = \mathbf{R}_\text{retr}$ overweights observations and produces overconfident posteriors.
• Block-diagonal across overpasses; cross-time correlation only within met decorrelation scale.

Prior on $S$ — spatially correlated, sign-constrained¶

Same structure as Tier II:

Adjoint¶

The adjoint of the transport equation (backward in time, conservative form) is:

The $\nabla\cdot(\mathbf{u}\,\lambda)$ term is conservative (matches the forward $\nabla\cdot(\mathbf{u}\,c)$); the doc previously had $\mathbf{u}\cdot\nabla\lambda$, which is equivalent only for divergence-free $\mathbf{u}$ and is incorrect for compressible WRF winds. JAX computes the discretised adjoint exactly via reverse-mode autodiff through the FV solver — no hand-derived adjoint code, no separate adjoint-correctness derivation. The mathematical form above is for reading, not implementation.

Incremental 4D-Var (the operational default)¶

Linearise around the current iterate $S^k$, solve the linear inner minimisation, update outer iterate:

$J_\text{lin}$ uses the tangent linear of the FV solver, trivially built via jax.linearize. Inner solves are quadratic in $\delta S$ → CG or Lanczos via gaussx. Cuts cost by 1–2 orders of magnitude vs. fully-nonlinear 4D-Var. This is the default; full nonlinear is a sanity check.

Control-variable transform¶

Direct optimisation in $S$-space with $\mathbf{B}^{-1}$ is infeasible — $\mathbf{B}$ for a $200 \times 200$ grid is $40000^{2} \approx 10^{9}$ entries. Standard fix:

$\mathbf{B}^{-1/2}$ materialised via Matérn factorisation in gaussx (Kronecker structure for separable correlation). This is the load-bearing trick that makes 4D-Var tractable; should be explicit in assimilation/control.py and on the API.

Posterior covariance — three options¶

4D-Var alone gives MAP; downstream Tier V needs the posterior:

• Gauss–Newton Hessian inversion. $\mathbf{P}^{*} = (\mathbf{H}^{\top} \mathbf{R}^{-1} \mathbf{H} + \mathbf{B}^{-1})^{-1}$ evaluated at MAP. Tractable for moderate grids via gaussx Krylov.
• Laplace around MAP. Sample from $\mathcal{N}(S^{*}, \mathbf{P}^{*})$; cheapest path.
• En4D-Var. Ensemble around MAP gives sample covariance; couples to filterax. Best when the posterior is non-Gaussian.

The posterior export to Tier V.A is via the same adapter pattern as Tier I/II — see the posterior-export module.

Cost / performance¶

For a $200 \times 200$ grid, 24-hour assimilation window, with incremental 4D-Var: ~minutes per outer iteration on CPU, seconds on GPU. Inner CG iteration is $O(n_\text{iter} \times n_\text{steps} \times n_\text{grid})$.

(3) Model emulator¶

Full 3D FV transport is expensive: $O(N^{3})$ state, repeated time integration. Emulator is essential here, not optional like at Tier I.

State variable — commit to 2D column for v1¶

Most satellite inversion needs only the column-integrated XCH₄, not the 3D field. Default emulator state is 2D column $c_\text{col}(x, y, t)$; full 3D is a v2 escalation when multi-layer in-situ assimilation enters scope. Order-of-magnitude difference in training cost.

Architecture options¶

• CNN UNet. Honest baseline for non-stationary terrain. No translation-invariance assumption.
• Graph-network operator. Right structural fit for unstructured analysis grids and basin-scoped problems.
• Fourier Neural Operator (FNO). Resolution-agnostic and natural fit with spectraldiffx philosophy — but assumes translation invariance in the kernel. Works well for periodic / homogeneous problems; breaks for urban basins with terrain. Use only when translation invariance is plausible.
• Neural ODE. $c(t+\Delta t) = f_\theta(c(t), \mathbf{u}(t))$ iterated. Simple but suffers from long-horizon drift.
• Reduced-order model (ROM). POD on simulation snapshots → low-rank basis; learn projected dynamics via Galerkin or DMD. Great when dynamics live on a low-dim manifold.

Training data¶

• Active-learning over uniform climatology binning. Sample WRF / ERA5 climatology adaptively — the emulator’s residual error map drives where to run the next FV simulation. Reaches operational accuracy with 100–300 runs vs. ~1000 for uniform sampling.
• Sample met conditions from the operational distribution (facility locations of interest, overpass times) rather than a uniform-bin climatology — same critique as Tier II’s emulator.

Emulator adjoint must be calibrated¶

Backprop through a trained emulator gives some gradient — whether it matches the true PDE adjoint is empirical.

(4) Emulator-based inference¶

Replace the FV integrator with the FNO / neural ODE in the 4D-Var loop:

• Gradient via autodiff through the emulator (not the PDE solver). Both are jax.grad-able; only the cost-per-iteration changes.
• Orders of magnitude faster per iteration (typically 100–1000×).
• Adjoint validation: emulator gradient ≈ FV adjoint (Step 3 calibration test).
• Posterior validation: posterior from emulator-based 4D-Var ≈ posterior from adjoint 4D-Var on the same observations. If they don’t agree, the emulator is biased — diagnose before trusting downstream.

(5) Amortized inference (predictor)¶

Output grid commitment¶

$S(\mathbf{x},t)$ is on the inversion grid (~1–10 km, basin-scoped). Predictor outputs a fixed-size 2D field per basin tile per time slice. Per-instrument predictor heads, dispatched by instrument_id — same pattern as Tier II.

Irregular temporal sampling¶

Satellite passes are intermittent (1–3 per day per instrument over a basin) with gaps. Input is not regularly sampled — $t_1, \dots, t_n$ are observation times, not a fixed grid. Architecture: set-transformer with time-encoding over irregular passes, fused with a regular-grid encoder over $\mathbf{u}_{1:T}$ from the met field. Vanilla seq2seq / ConvLSTM assumes regular sampling and will silently underperform.

Posterior over the spatial source field¶

Conditional flow over images vs. score-based diffusion — same trade-off as Tier II:

• gauss_flows is currently 1D-only; 2D coupling layers are a multi-month extension.
• Score-based diffusion is the safer v1 path for the spatial posterior.
• Context conditioning via FiLM / hypernet primitives in pyrox.nn — same pattern as Tier I/II.

(6) Improve¶

• Multi-species coupling. Add CO and CO₂ tracers; their source ratios constrain CH₄ source attribution (e.g. fossil vs. agricultural).
• Adaptive grid refinement. Refine near sources, coarsen elsewhere. finitevolX may need primitives for this.
• Online DA. Assimilate observations as they arrive (rolling-window 4D-Var, or filterax’s ensemble Kalman smoother).
• Sub-grid plume reconstruction. When a source is sub-grid, the FV solver smears it; pair Tier III with a Tier I plume in the near-field for better point-source representation.
• Hierarchical Matérn length-scale. Promote $\ell$ to a hyperparameter with its own posterior — let the data choose the regularisation scale (mirrors Tier II).

Module layout¶

Validation strategy¶

• 1D diffusion. Initial Dirac → at time $t$, solution is Gaussian with variance $2Kt$. Check $L^2$ error vs. analytical.
• 1D advection. Cosine pulse advected at constant $u$ → after one period, recover initial condition. Tests upwind / flux-limiter consistency.
• CFL stability. Artificially exceed CFL → expect blow-up; stay within → expect stability. Catches silent flux-limiter regressions.
• Mass conservation — three regimes.
  • No sources, periodic BCs: $\int c\, \mathrm{d}V$ exact to floating-point.
  • With sources, no deposition: $\Delta \int c\, \mathrm{d}V = \iint S\, \mathrm{d}V\, \mathrm{d}t$ summed over the window.
  • With deposition: include $-\lambda \int c\, \mathrm{d}V\, \mathrm{d}t$ term.
• Adjoint correctness. JAX vjp of the discrete forward should satisfy $\langle \mathbf{F}\mathbf{u}, \mathbf{v}\rangle = \langle \mathbf{u}, \mathbf{F}^{\top} \mathbf{v}\rangle$ for random $\mathbf{u}, \mathbf{v}$. Cheap, catches differentiation bugs.
• Tier I limit. For a single point source in a uniform wind with constant $\mathbf{K}$, the steady-state FV solution should match the Gaussian-plume formula at downwind distances much greater than the grid spacing.
• Emulator-adjoint calibration. Emulator-autodiff gradients vs. FV-autodiff gradients on a held-out met set, $<5\%$ relative error in operator norm. Hard test — failure means Step 4 inversion is biased.
• Emulator OOD generalization. Train on one met regime (e.g. summer Permian), evaluate on another (winter Permian, or a different basin). Resists overfit-to-training-distribution.
• Real-data benchmark. Compare 4D-Var output to existing CAMS / GEOS-Chem-Adjoint inversions on a published time window (e.g. Maasakkers et al., 2023Jacob et al., 2022 Permian inversions). Posterior credible interval should overlap the published estimate. Without this, the inversion is a synthetic exercise.

Open questions¶