`plumax` equation registry Offline bookkeeping index of every labelled equation across the roadmap
plumax — equation registry¶ Developer aid. This file is a bookkeeping index of every labelled equation across the plumax roadmap notes. It exists so that authors can sanity-check label uniqueness, plan future cross-references, and audit equation numbering at a glance.
Scope: all {math} blocks under roadmap/Convention: every block label is eq-<page>-<short-name>. <page> matches the file basename (tier1, tier2, …) or sub-page tag (va, vb, vc, vd, rtm, prereqs).
Total: 77 labelled equations across 11 pages.
Label Form Section eq-tier1-gaussian-plumeSteady-state Gaussian plume with PBL image-source sum Gaussian plume eq-tier1-gaussian-puffSum of 3D Gaussian puffs advected by the wind Gaussian puff eq-tier1-column-aky_model = A (∫c dz + c_bg) — column + AK forwardColumn + AK eq-tier1-likelihoodHeteroscedastic Gaussian observation likelihood Likelihood eq-tier1-amortizedNPE / amortized predictor signature Amortized inference eq-tier1-pbl-wellmixedVertically well-mixed limit c → Q / (ū L √(2π) σ_y) exp(−y²/2σ_y²) Validation — PBL capping
Label Form Section eq-tier2-langevinMarkov-1 Langevin SDE on particle velocity + position Langevin dynamics eq-tier2-dt-boundAdaptive timestep bound from τ L \tau_L τ L Time-stepping eq-tier2-footprintBackward-mode source–receptor footprint integral Footprint definition eq-tier2-forwardy = A col_z(F q) + c_bg + εForward observation operator eq-tier2-likelihoodε ~ N(0, R_retr + R_repr)Likelihood eq-tier2-posterior-meanLognormal posterior mean — Gaussian–Gaussian linearised Closed form eq-tier2-posterior-covLognormal posterior covariance same eq-tier2-amortizedPredictor signature (y, met) → p(log q(x)) Amortized inference
Label Form Section eq-tier3-conservationρ \rho ρ Conservation eq. eq-tier3-forwardy_t = A_t col_z(c(S, c₀, t)) + c_bg + ε_tForward operator eq-tier3-4dvar-cost3-term 4D-Var cost (source + IC + obs sum) 4D-Var cost eq-tier3-likelihoodPer-time heteroscedastic Gaussian Likelihood eq-tier3-adjointConservative adjoint transport equation Adjoint eq-tier3-incrementalIncremental 4D-Var outer-iterate update Incremental 4D-Var eq-tier3-control-xformχ = B − 1 / 2 ( S − S b ) \boldsymbol{\chi} = \mathbf{B}^{-1/2}(S − S_b) χ = B − 1/2 ( S − S b ) Control transform eq-tier3-amortizedSequence predictor signature Amortized inference
Label Form Section eq-rtm-radianceSWIR Beer–Lambert radianceBeer–Lambert eq-rtm-tau-totalTwo-way airmass-factor optical depth same eq-rtm-tauLayer-integrated τ \tau τ same eq-rtm-thermal-irTIR variant (emission + path radiance) Thermal-IR addendum eq-rtm-state-vectorJoint retrieval state vector Joint state eq-rtm-gauss-newtonIterative Gauss–Newton update Gauss–Newton eq-rtm-gainGain matrix G k \mathbf{G}_k G k same eq-rtm-convergenceRodgers (2000) §5.7 convergence criterion same eq-rtm-posterior-covS ∗ \mathbf{S}^* S ∗ Info content eq-rtm-averaging-kernelA = G K \mathbf{A} = \mathbf{G}\mathbf{K} A = GK same eq-rtm-info-contentShannon H H H Δ S \Delta\mathbf{S} Δ S same eq-rtm-factorised-lutFactorised LUT (gas × surf + scatt) decomposition Factorised LUT eq-rtm-jax-gradjax.grad ‖NeuralRTM(x) − y‖² end-to-end gradientEmulator-based inference eq-rtm-amortizedDirect-retrieval predictor signature Amortized inference
Label Form Section eq-tier4-per-inst-forwardPer-instrument coupled forward Per-instrument forward eq-tier4-per-inst-likelihoodPer-instrument noise covariance decomposition same eq-tier4-state-vectorFull coupled state vector enumeration State vector eq-tier4-fused-operatorJoint multi-instrument observation operator Multi-instrument fusion eq-tier4-q-ouOU process for Q ( t ) Q(t) Q ( t ) Q ( t ) Q(t) Q ( t ) eq-tier4-gradientEnd-to-end multi-instrument gradient Gradient eq-tier4-cost3-term coupled 4D-Var cost (per-inst obs + prior + Q ( t ) Q(t) Q ( t ) Cost function eq-tier4-coupled-emulatorCoupled emulator g ϕ g_\phi g ϕ Coupled emulator eq-tier4-amortizedOperational predictor signature Amortized inference
Label Form Section eq-tier5-loglikTMTPP three-term log-likelihood Log-likelihood eq-tier5-markPer-event mark integral ∫ P d L i f d Q \int P_d L_i f \, \mathrm{d}Q ∫ P d L i f d Q Mark contribution eq-tier5-importance-weightsImportance-weighted MC estimator f / π per-event f / \pi_\text{per-event} f / π per-event same
Label Form Section eq-va-full-posteriorPer-event posterior ∝ L i ⋅ π per-event \propto L_i \cdot \pi_\text{per-event} ∝ L i ⋅ π per-event Full-physics posterior eq-va-wind-rescaleWind-source consistency rescaling Catalog Q eq-va-mark-integralMark-likelihood integral form Mark likelihood eq-va-importance-mcImportance-weighted MC estimator same eq-va-thinned-rateIntegrated thinned-rate term Detection-floor / non-detection eq-va-multisourceMulti-source product over within-overpass sources Multi-source
Label Form Section eq-vb-lambda-constantHomogeneous Poisson Temporal eq-vb-lambda-diurnalDiurnal sinusoidal intensity same eq-vb-lambda-stepStep intensity (valve fail) same eq-vb-lambda-decayExponential decay (blowdown) same eq-vb-lambda-hawkesHawkes / self-exciting kernel same eq-vb-lambda-lgcpLog-Gaussian Cox process same eq-vb-pod-hillHill function P d ( Q ) = 1 / ( 1 + ( Q 50 / Q ) k ) P_d(Q) = 1/(1 + (Q_{50}/Q)^k) P d ( Q ) = 1/ ( 1 + ( Q 50 / Q ) k ) POD eq-vb-pod-hier-priorHierarchical prior on ( Q 50 , k ) (Q_{50}, k) ( Q 50 , k ) POD calibration eq-vb-loglikCanonical TMTPP log-likelihood Likelihood — canonical eq-vb-mark-iwImportance-weighted MC for the mark integral Practical evaluation eq-vb-loglik-pointPoint-regime simplification Point regime
Label Form Section eq-vc-wait-homogeneousE [ Δ t ] = 1 / λ 0 \mathbb{E}[\Delta t] = 1/\lambda_0 E [ Δ t ] = 1/ λ 0 Wait time eq-vc-wait-inhomogeneousInhomogeneous-Poisson wait integral same eq-vc-occurrence-homogeneous1 − e − λ 0 Δ t 1 - e^{-\lambda_0 \Delta t} 1 − e − λ 0 Δ t Occurrence eq-vc-occurrence-inhomogeneous1 − e − ∫ λ d t 1 - e^{-\int \lambda \, \mathrm{d}t} 1 − e − ∫ λ d t same eq-vc-hawkes-bumpConditional intensity given prior detection Conditional intensity eq-vc-cumulativeE [ N ( 0 , T ) ] = ∫ 0 T λ d t \mathbb{E}[N(0,T)] = \int_0^T \lambda \, \mathrm{d}t E [ N ( 0 , T )] = ∫ 0 T λ d t Cumulative count
Label Form Section eq-vd-mtotalM total ( T ) = E [ N true ] ⋅ E [ Q ] M_\text{total}(T) = \mathbb{E}[N_\text{true}] \cdot \mathbb{E}[Q] M total ( T ) = E [ N true ] ⋅ E [ Q ] Corrected estimator eq-vd-mnaiveBiased naive estimator (sum over detected) same eq-vd-ndetectedE [ N detected ] \mathbb{E}[N_\text{detected}] E [ N detected ] same eq-vd-mtotal-posteriorPosterior samples for M total M_\text{total} M total Posterior eq-vd-pod-unionUnion POD across K K K Multi-satellite fusion
Maintenance ¶ When you add or remove a {math} block:
Add / remove the corresponding row in this file.
Re-run the count grep -rE "^:label: eq-" projects/plume_simulation/notes/ | wc -l and update the total at the top.
If a label changes, grep for {eq} references across the roadmap (grep -rE "\{eq\}\`" projects/plume_simulation/notes/`) and rename them too.