Equation index

Design-roadmap equation index¶

An index of the key equations across the design roadmap pages, grouped by page, with a link to the section where each one lives. It exists so that authors can plan cross-references and audit coverage without reading every page.

00 — Prerequisites — `design/00_prerequisites.md`¶

Label	Form	Section
`eq-ak-operator`	`ŷ = A (h^T x + (1−h^T) x_a)`	Averaging-kernel operator

I — Tier I (Gaussian family) — `design/01_tier1_gaussian.md`¶

Label	Form	Section
`eq-tier1-gaussian-plume`	Steady-state Gaussian plume with PBL image-source sum	Gaussian plume
`eq-tier1-gaussian-puff`	Sum of 3D Gaussian puffs advected by the wind	Gaussian puff
`eq-tier1-column-ak`	`y_model = A (∫c dz + c_bg)` — column + AK forward	Column + AK
`eq-tier1-likelihood`	Heteroscedastic Gaussian observation likelihood	Likelihood
`eq-tier1-amortized`	NPE / amortized predictor signature	Amortized inference
`eq-tier1-pbl-wellmixed`	Vertically well-mixed limit `c → Q / (ū L √(2π) σ_y) exp(−y²/2σ_y²)`	Validation — PBL capping

II — Tier II (Lagrangian particle) — `design/02_tier2_lagrangian.md`¶

Label	Form	Section
`eq-tier2-langevin`	Markov-1 Langevin SDE on particle velocity + position	Langevin dynamics
`eq-tier2-dt-bound`	Adaptive timestep bound from $\tau_L$ , CFL, diffusion	Time-stepping
`eq-tier2-footprint`	Backward-mode source–receptor footprint integral	Footprint definition
`eq-tier2-forward`	`y = A col_z(F q) + c_bg + ε`	Forward observation operator
`eq-tier2-likelihood`	`ε ~ N(0, R_retr + R_repr)`	Likelihood
`eq-tier2-posterior-mean`	Lognormal posterior mean — Gaussian–Gaussian linearised	Closed form
`eq-tier2-posterior-cov`	Lognormal posterior covariance	same
`eq-tier2-amortized`	Predictor signature `(y, met) → p(log q(x))`	Amortized inference

III — Tier III (Eulerian FV) — `design/03_tier3_eulerian.md`¶

Label	Form	Section
`eq-tier3-conservation`	$\rho$ -weighted advection–diffusion–source–sink	Conservation eq.
`eq-tier3-forward`	`y_t = A_t col_z(c(S, c₀, t)) + c_bg + ε_t`	Forward operator
`eq-tier3-4dvar-cost`	3-term 4D-Var cost (source + IC + obs sum)	4D-Var cost
`eq-tier3-likelihood`	Per-time heteroscedastic Gaussian	Likelihood
`eq-tier3-adjoint`	Conservative adjoint transport equation	Adjoint
`eq-tier3-incremental`	Incremental 4D-Var outer-iterate update	Incremental 4D-Var
`eq-tier3-control-xform`	$\boldsymbol{\chi} = \mathbf{B}^{-1/2}(S − S_b)$ control transform	Control transform
`eq-tier3-amortized`	Sequence predictor signature	Amortized inference

RTM — RTM stack — `design/04_rtm_stack.md`¶

Label	Form	Section
`eq-rtm-radiance`	SWIR Beer–Lambert radiance	Beer–Lambert
`eq-rtm-tau-total`	Two-way airmass-factor optical depth	same
`eq-rtm-tau`	Layer-integrated $\tau$ from HAPI cross-section	same
`eq-rtm-thermal-ir`	TIR variant (emission + path radiance)	Thermal-IR addendum
`eq-rtm-state-vector`	Joint retrieval state vector	Joint state
`eq-rtm-gauss-newton`	Iterative Gauss–Newton update	Gauss–Newton
`eq-rtm-gain`	Gain matrix $\mathbf{G}_k$ + Jacobian definition	same
`eq-rtm-convergence`	Rodgers (2000) §5.7 convergence criterion	same
`eq-rtm-posterior-cov`	$\mathbf{S}^*$ posterior covariance	Info content
`eq-rtm-averaging-kernel`	$\mathbf{A} = \mathbf{G}\mathbf{K}$ , DOFs = tr	same
`eq-rtm-info-content`	Shannon $H$ + posterior contraction $\Delta\mathbf{S}$	same
`eq-rtm-factorised-lut`	Factorised LUT (gas × surf + scatt) decomposition	Factorised LUT
`eq-rtm-jax-grad`	`jax.grad ‖NeuralRTM(x) − y‖²` end-to-end gradient	Emulator-based inference
`eq-rtm-amortized`	Direct-retrieval predictor signature	Amortized inference

IV — Tier IV (Coupled E2E) — `design/05_tier4_coupled.md`¶

Label	Form	Section
`eq-tier4-per-inst-forward`	Per-instrument coupled forward	Per-instrument forward
`eq-tier4-per-inst-likelihood`	Per-instrument noise covariance decomposition	same
`eq-tier4-state-vector`	Full coupled state vector enumeration	State vector
`eq-tier4-fused-operator`	Joint multi-instrument observation operator	Multi-instrument fusion
`eq-tier4-q-ou`	OU process for $Q(t)$	$Q(t)$
`eq-tier4-gradient`	End-to-end multi-instrument gradient	Gradient
`eq-tier4-cost`	3-term coupled 4D-Var cost (per-inst obs + prior + $Q(t)$ kernel)	Cost function
`eq-tier4-coupled-emulator`	Coupled emulator $g_\phi$ signature	Coupled emulator
`eq-tier4-amortized`	Operational predictor signature	Amortized inference

V — Tier V index — `design/06_tier5_population.md`¶

Label	Form	Section
`eq-tier5-loglik`	TMTPP three-term log-likelihood	Log-likelihood
`eq-tier5-mark`	Per-event mark integral $\int P_d L_i f \, \mathrm{d}Q$	Mark contribution
`eq-tier5-importance-weights`	Importance-weighted MC estimator $f / \pi_\text{per-event}$	same

V.A — Instantaneous emission — `design/06_tier5a_instantaneous.md`¶

Label	Form	Section
`eq-va-full-posterior`	Per-event posterior $\propto L_i \cdot \pi_\text{per-event}$	Full-physics posterior
`eq-va-wind-rescale`	Wind-source consistency rescaling	Catalog Q
`eq-va-mark-integral`	Mark-likelihood integral form	Mark likelihood
`eq-va-importance-mc`	Importance-weighted MC estimator	same
`eq-va-thinned-rate`	Integrated thinned-rate term	Detection-floor / non-detection
`eq-va-multisource`	Multi-source product over within-overpass sources	Multi-source

V.B — TMTPP — `design/06_tier5b_point_process.md`¶

Label	Form	Section
`eq-vb-lambda-constant`	Homogeneous Poisson	Temporal
`eq-vb-lambda-diurnal`	Diurnal sinusoidal intensity	same
`eq-vb-lambda-step`	Step intensity (valve fail)	same
`eq-vb-lambda-decay`	Exponential decay (blowdown)	same
`eq-vb-lambda-hawkes`	Hawkes / self-exciting kernel	same
`eq-vb-lambda-lgcp`	Log-Gaussian Cox process	same
`eq-vb-pod-hill`	Hill function $P_d(Q) = 1/(1 + (Q_{50}/Q)^k)$	POD
`eq-vb-pod-hier-prior`	Hierarchical prior on $(Q_{50}, k)$	POD calibration
`eq-vb-loglik`	Canonical TMTPP log-likelihood	Likelihood — canonical
`eq-vb-mark-iw`	Importance-weighted MC for the mark integral	Practical evaluation
`eq-vb-loglik-point`	Point-regime simplification	Point regime

V.C — Persistency — `design/06_tier5c_persistency.md`¶

Label	Form	Section
`eq-vc-wait-homogeneous`	$\mathbb{E}[\Delta t] = 1/\lambda_0$	Wait time
`eq-vc-wait-inhomogeneous`	Inhomogeneous-Poisson wait integral	same
`eq-vc-occurrence-homogeneous`	$1 - e^{-\lambda_0 \Delta t}$	Occurrence
`eq-vc-occurrence-inhomogeneous`	$1 - e^{-\int \lambda \, \mathrm{d}t}$	same
`eq-vc-hawkes-bump`	Conditional intensity given prior detection	Conditional intensity
`eq-vc-cumulative`	$\mathbb{E}[N(0,T)] = \int_0^T \lambda \, \mathrm{d}t$	Cumulative count

V.D — Total emission — `design/06_tier5d_total_emission.md`¶

Label	Form	Section
`eq-vd-mtotal`	$M_\text{total}(T) = \mathbb{E}[N_\text{true}] \cdot \mathbb{E}[Q]$	Corrected estimator
`eq-vd-mnaive`	Biased naive estimator (sum over detected)	same
`eq-vd-ndetected`	$\mathbb{E}[N_\text{detected}]$ formula	same
`eq-vd-mtotal-posterior`	Posterior samples for $M_\text{total}$	Posterior
`eq-vd-pod-union`	Union POD across $K$ satellites	Multi-satellite fusion

Maintenance¶

When you add, remove or label an equation on a design page:

Add / remove the corresponding row in this file.
Check label uniqueness across the whole docs tree: grep -rhoE "$eq-[a-z0-9-]+$" docs/ --include=*.md | sort | uniq -d.
If a label changes, rename every [text](#old-label) and {eq} reference to it as well.

Equation index

Design-roadmap equation index¶

00 — Prerequisites — design/00_prerequisites.md¶

I — Tier I (Gaussian family) — design/01_tier1_gaussian.md¶

II — Tier II (Lagrangian particle) — design/02_tier2_lagrangian.md¶

III — Tier III (Eulerian FV) — design/03_tier3_eulerian.md¶

RTM — RTM stack — design/04_rtm_stack.md¶

IV — Tier IV (Coupled E2E) — design/05_tier4_coupled.md¶

V — Tier V index — design/06_tier5_population.md¶

V.A — Instantaneous emission — design/06_tier5a_instantaneous.md¶

V.B — TMTPP — design/06_tier5b_point_process.md¶

V.C — Persistency — design/06_tier5c_persistency.md¶

V.D — Total emission — design/06_tier5d_total_emission.md¶