# Scalar time-dependent transport: specialist derivation and verification This note describes the executed R4 finite-cylinder model and what the light review independently recomputes. It is not a new physical validation. The tabulated-GHM point-nadir estimator remains nonuniformly converged. This is an expected warning, not a pass condition to suppress. The raw ice LUT is deliberately absent. ## Symbols and units | Symbol | Meaning | SI units | |---|---|---| | \(L(\mathbf r,\hat\Omega,t)\) | photon radiance propagating along \(\hat\Omega\) | photons m⁻² s⁻¹ sr⁻¹ | | \(\Phi_s(t),N_s\) | emitted photon rate and event photon count | photons s⁻¹; photons | | \(\beta_e,\beta_s,\beta_a\) | extinction, scattering, absorption coefficients | m⁻¹ | | \(\omega=\beta_s/\beta_e\) | single-scattering albedo | dimensionless | | \(p(\mu),f(\mu)\) | phase density per solid angle; density per cosine | sr⁻¹; dimensionless per unit \(\mu\) | | \(g\) | mean scattering cosine | dimensionless | | \(A,\mathbf n_A\) | small planar collecting area and normal toward scene | m²; unit vector | | \(R_0,r_j,\ell_j\) | direct range, collision-observer range, accumulated source-collision path | m | | \(K(\Delta),h(\Delta)\) | aperture-area transfer response and vacuum-normalized response | m⁻² s⁻¹; s⁻¹ | | \(m_{bi}\) | relative response MASS in time bin i of batch b | dimensionless | | \(\delta t,B,N_b\) | native bin width, independent batches, histories per batch | s; counts; counts | ## RTE and boundary/source statement For scalar, elastic, stationary-medium transport: \[ \tag{R1} \left(c^{-1}\partial_t+\hat\Omega\!\cdot\!\nabla\right)L =-\beta_e L+\beta_s\int_{4\pi}p(\hat\Omega'\!\cdot\!\hat\Omega)L(\mathbf r,\hat\Omega',t)\,d\Omega' +\frac{\Phi_s(t)}{4\pi}\delta^3(\mathbf r-\mathbf r_s). \] The isotropic point source emits into all 4π. Coefficients are constant inside the cylinder and zero outside. There is no incoming exterior illumination, ground reflection or reflecting cloud boundary. Exterior photons move in vacuum. The cylinder is convex, so a ray that has left cannot re-enter without a change of direction outside, which this model excludes. An above-top downward launch may enter the cloud before leaving again. The main grid assumes \(\omega=1\), hence \(\beta_a=0\). This is a model input, not measured UV ice absorption. There is no molecular scattering, ozone, aerosol, refraction or Earth curvature. A full physical atmosphere would need additional spatially varying scattering/absorption terms and their path delays. Cloud base 14 km, top 17 km and radius 30 km are example geometry. On-axis sources are 15/16/17/18 km; the observer is 550 km and has a 60 degree full circular FOV. The area response uses a flat horizontal aperture. A finite source, real cloud structure and off-axis population are not represented. ## Phase conventions and actual input \[ \tag{P1} \int_{4\pi}p\,d\Omega=1,\qquad f(\mu)=2\pi p(\mu),\qquad \int_{-1}^{1}f(\mu)\,d\mu=1,\quad g=\int_{-1}^{1}\mu f(\mu)\,d\mu. \] Azimuth is uniform for the randomly oriented scalar model. The supplied P11 convention has integral2 in cosine, so normalizing its sampled representation and dividing f by 2π supplies the probability per steradian. P11 itself must not be used as p without normalization. Native330/340 nm P11 at effective radius 10 μm is linearly interpolated in wavelength to the numerical coordinate 337.1 nm. The resulting density is represented piecewise linearly in \(\mu\), normalized, and its inverse CDF solved segment by segment. Sampling and next-event evaluation use this identical representation. Its computed \(g=0.7709446878541932\). The HG comparison is \[ \tag{P2} f_{HG}(\mu)=\frac{1-g^2}{2(1+g^2-2g\mu)^{3/2}},\qquad \langle P_2(\mu)\rangle_{HG}=g^2. \] The review independently integrates HG normalization/moments, checks its inverse CDF, and draws seeded angles. It does **not** pretend to recompute GHM normalization without the raw LUT. The processed ice-habit mixture, roughness, size-distribution applicability, interpolation and omitted polarization remain physical input uncertainties. The molecular band and air/vacuum wavelength registration are not folded here. Raw dataset: https://docs.hygeos.com/public.php/webdav/ic_baum_ghm_sol.nc ; developer information: https://www.ssec.wisc.edu/ice_models/polarization.html . Original full NetCDF SHA256 is 2b37e27cffbbf0dd7e048df16c5fb612cd7a83e3c657c851bb50e758e11e870e; executed private bracket SHA256 is 2cb7bca802632479473a0812912ec8007fc69087ade2767d3c06cc3e7cb81a02. Rights to redistribute the raw LUT were not established. A reproducible private extraction recipe is provided in the separate R4 code bundle; obtaining that input is an optional prerequisite for a full RT rerun, not for these light checks. ## Direct contribution: deterministic and independently testable For the on-axis vertical source-observer segment, define \(H=z_t-z_b\): \[ \tag{D1} \beta_e=\tau_{total}/H,\quad \ell_{above}=\max[0,\min(z_t-z_s,H)],\quad \tau_{above}=\beta_e\ell_{above}. \] \[ \tag{D2} K_{vac}=\frac1{4\pi R_0^2},\qquad K_0=K_{vac}e^{-\tau_{above}},\quad R_0=z_o-z_s. \] K0 has units m⁻² per emitted photon; the exact unscattered delay response is \(K_0\delta(\Delta)\). For an off-axis source the receiver cosine and actual finite cloud-segment length must be included; D2 is this grid's on-axis specialization. The review recomputes D1/D2 for all 56 cases from geometry and total tau, not by copying the stored direct value. At 15 km overlying tau is 2/3 of total tau; at 16 km it is 1/3; at/above 17 km it is 0. Varying total tau changes cloud below an internal source as well as above it. Thus this sweep is not a universal relation in overlying tau alone. The total directional response may exceed vacuum because downward emission is redirected upward; it is not an escape probability and need not be at most1. ## Next-event estimator and source weights The physical source angular density is \(q_0=1/(4\pi)\). If an initial direction is sampled from another normalized density q, use \[ \tag{N1}w_{launch}=q_0/q. \] The production internal-source proposal mixes isotropic and zenith-centered phase densities with equal weights. Its likelihood weight is bounded above by2. This affects launch efficiency only; physical scatter directions retain the chosen phase law. At/above the cloud top the production launch is isotropic. The true source remains isotropic in every case. For collision j, \(\hat u_j=(\mathbf r_o-\mathbf r_j)/r_j\) points **toward** the observer and \(\mu_j=\hat\Omega_{in,j}\cdot\hat u_j\). With a receiver normal toward the scene, \[ \tag{N2}c_{A,j}=\max(0,-\mathbf n_A\!\cdot\!\hat u_j). \] A small receiver subtends \(A c_{A,j}/r_j^2\). Let \(\chi_{FOV}\) be its circular acceptance. If wj is weight **before** implicit absorption at collision j, \[ \tag{N3} \widehat K_{scat}=\frac1{N_b}\sum_{history}\sum_j w_j\omega\,p(\mu_j)\frac{c_{A,j}}{r_j^2} \exp[-\beta_e\ell(\mathbf r_j\to\mathbf r_o)]\chi_{FOV,j}. \] This is per unit area. Multiply by A once to obtain a collected fraction. The segment length in the exponential is the intersection of the **finite collision-observer segment** with the cloud, not an infinite ray or a vertical shortcut. In the main conservative case omega 1 contributes no extra factor. In an implicit-absorption implementation that has already multiplied w by omega before evaluating N3, omit the displayed additional omega. Double counting it would be wrong. The direct contribution D2 is added exactly once per emitted history ensemble, not at every collision. A source-only proposal needs no extra 4π multiplier beyond N1. A phase function per steradian and its cosine density are not interchangeable. Aperture cosine, area and QE/PDE also must not be counted twice; QE/PDE is absent from this transport calculation. ## Physical delays, impulse atom and native histogram \[ \tag{T1}\Delta_j=(\ell_j+r_j-R_0)/c\ge0,\qquad t_{direct}=R_0/c. \] The accumulated physical path includes an external source's vacuum travel to cloud entry. Boundary numerical tolerances must not create physical path or delay. The triangle inequality supplies causality; a diffusion PDE is a different approximation and can have pre-flight tails. Let \(h=K/K_{vac}\). For native bin i, the stored value is \[ \tag{T2}m_{bi}=\int_{i\delta t}^{(i+1)\delta t}h_{scat,b}(\Delta)\,d\Delta. \] The NPZ arrays contain scattered **mass**, not density, and a final overflow element for delays at/after 12 ms. Their ordinary bins are 2 μs wide. Mean density is mass divided by bin width in seconds. The direct atom has relative mass \(e^{-\tau_{above}}\), supplied separately. In the reviewed convolution it is at0, while ordinary scattered mass is placed at bin centers 1,3,5,… μs on a 1 μs grid. Plot bins later coarsen only the tail. The native arrays, rather than plotted/rebinned arrays, are used for this review's fluence reconciliation. Zero observed overflow is an empirical statement about these histories, not proof that every possible path arrives within 12 ms. Truncated packet weight is separately recorded. No production packet reached the 20000 scatter safety cap. ## Pulse convolution, retarded time and rise definitions With retarded time \(u=t_{absolute}-t_{source,onset}-R_0/c\), \[ \tag{T3}\frac{\dot N_{ap}(u)}A=\int_0^\infty K(\Delta)\Phi_s(u-\Delta)\,d\Delta. \] The assumed unit-peak source shape is \[ \tag{T4}s(u)=C_s e^{-u/\tau_d}(1-e^{-u/\tau_r})H(u),\quad \tau_d=3\,ms,\quad u_{peak}=\tau_r\ln(1+\tau_d/\tau_r). \] Cs normalizes the source's own peak to1. tau_r is calibrated to intrinsic rising 10–90% times 30/80/200 μs, relative to that peak. The 3 ms is an assumed exponential decay time; it is not a fitted 3 ms duration or a measured intrinsic NBE waveform. In a lossy/broadened pulse, renormalizing each plotted curve to its own peak is useful for shape only; compare the separately stored peak gain/fluence for photometry. An optional ideal first-order response is \(e(u)=2\pi f_c e^{-2\pi f_cu}H(u)\), with fc0 representing a delta. The mean waveform is \(s*h*e\). Analog cutoff is not sample rate, and this does not simulate actual post-counting noise correlations or a complete anti-alias chain. The reported metrics are separately \(t_{90}-t_{10}\) and \(t_{peak}-t_{onset}\). Onset is the known model onset 0. A vanishingly weak direct ray can define that mathematical onset even when it is unobservable. Heritage observed rise numbers cannot be inserted as intrinsic10–90%parameters without a source/detector fit. No population classifier is validated here. ## Independent batch uncertainty and covariance For gain \(G_b=G_0+\sum_i m_{bi}\), with independent equal-size batches, \[ \tag{U1}\bar G=B^{-1}\sum_bG_b,\qquad SE(\bar G)=\sqrt{\frac{\sum_b(G_b-\bar G)^2}{B(B-1)}}. \] \[ \tag{U2}\operatorname{Var}(\bar G)=\sum_i\sum_j\operatorname{Cov}(\bar m_i,\bar m_j). \] The direct term is deterministic, so it adds no MC variance. A history can contribute to several bins/collisions. Adding only squared bin errors omits covariance and is generally wrong. The review computes the sum-covariance identity from the B×J centered matrix without allocating a J×J covariance matrix, and compares it with the stored fluence SE. It also displays the incorrect independent-bin estimate for contrast. Finite-batch SE is not a guaranteed confidence interval for an underresolved heavy-tail estimator. For the 15 km/tau 10 GHM control, increasing from 300000 to 1200000 histories changes the estimated gain noticeably. A 1 degree cone proposal preserves isotropic source moments but does not remove multi-scatter variance; some batches concentrate a large share of their estimate in a few histories. These are retained convergence failures, not evidence of a physical GHM/HG timing difference. Timing SEM from individual batch waveforms is a nonlinear diagnostic, especially for an argmax; it is not automatically the sampling SE of the timing functional of the averaged kernel. ## Executable equation-based first-scatter check For an isotropic phase, axisymmetry lets azimuth integrate out. At initial cosine mu, let se(mu) be the distance to the cylinder boundary. For a first collision at distance s, let r(s,mu) and ellout be its observer range and remaining cloud path. The relative first-order fluence is \[ \tag{S1}G^{(1)}=\frac{R_0^2}{2}\int_{-1}^{1}d\mu\int_0^{s_e(\mu)} \beta_e e^{-\beta_e[s+\ell_{out}(s,\mu)]}\frac{c_A(s,\mu)}{r(s,\mu)^2}\,ds. \] The review implements this integral independently with Gauss quadrature, splitting the cosine intervals where side and horizontal boundary paths exchange. Order 24/48/96 results and the retained isotropic MC first-collision estimate are compared. This is a first-collision equation check, not an independent full RT engine or a GHM check. ## What the review checks and does not check All56direct terms, native batch mass/gain/SE, nonnegative bins, overflow, packet-count accounting and weighted boundary/escape bookkeeping are recomputed from packaged inputs. The source-importance launch weights average1statistically and are not forcibly normalized. The review checks that escaped plus explicitly truncated weight equals launch weight, not that every finite importance sample sums exactlyN. Raw metadata reports no noncausal paths and no truncation. The individual three-dimensional histories are not retained, so those per-path geometry statements cannot be independently replayed from a histogram. The code, source input and seeds would be needed for replay. No original RT run is triggered by this notebook. Two earlier parameter-matched ideal-HG CloudScat comparisons agreed within their combined MC errors; see cloud_validation_summary.json. They test these ideal HG geometries only, use a shared timing postprocessor, and do not validate GHM, real atmosphere or a source-depth classifier. CloudScat primary method: https://gmd.copernicus.org/articles/13/5549/2020/ . The executable reviewer maps these equations to exact function names and source lines in reference/cloud and writes checks/transport_code_map.json. It records file hashes. The original executed engine/runner snapshots, when supplied, must be distinguished from later canonical files that added diagnostics. Do not infer byte-identical execution from a similar filename. Optional full transport reruns require the lawful raw bracket and the separate reproduction launcher. They are deliberately outside the default light notebook. Empirical cloud microphysics/structure, absorption/gases, real source spectrum and waveform, noisy onset and labelled-event inference remain open. ## Worked-history ledger The notebook also shows a fresh HG history with fixed seed2026102202,source16km,total tau10 and the same mean cosine as the main model. Six collisions occur before escape. Each row records launch/propagation weights,an exponential free optical path,the3D collision position,incoming and outgoing unit vectors,the phase probability toward nadir,observer optical depth and survival,range,aperture cosine,delay,and raw next-event summand. The ledger is an explanatory example,not an original production packet or a reported event-fluence estimate. Its end status distinguishes physical escape from a deliberate illustration cap.