The SBMP, MBSP and MBMP family

The radiative transfer equation in the SWIR¶

TL;DR. Two terms of the radiative transfer equation vanish in the shortwave infrared, and what is left integrates to Beer–Lambert.

Problem. For the intensity $I_\lambda$ in a plane-parallel atmosphere,

with $\mu = \cos\theta$, $\tau_\lambda$ the optical depth and $J_\lambda$ the source function, emission plus scattering into the beam. Between 1.6 and 2.3 μm both are small. Thermal emission: a 300 K surface emits $2\times10^{-4}$ of the sunlight it reflects at 2.2 μm (albedo 0.2, $\theta_s = 35°$). Rayleigh scattering: its optical depth falls as $\lambda^{-4}$, 16 times weaker at 2 μm than at 1 μm, and is $3.7\times10^{-4}$ in band 12. With $J_\lambda \to 0$, (1) reduces along the path $s$ to

with $\sigma_\lambda$ the absorption cross-section and $n$ the absorber’s density. Integrating (2) down to a Lambertian surface of albedo $A_\lambda$ and back up, through the column $\Omega = \int n\, dz$ along the slant paths, it gives the TOA reflectance

where $L_\lambda$ is the radiance at the sensor, $E_{\lambda,\odot}$ the solar irradiance and $m$ the geometric air-mass factor, as in radtrax · 1. Every pixel has two unknowns per wavelength, $A_\lambda$ and $\Omega$; the rest of the page is about removing $A_\lambda$.

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# Shapes: scalars; λ in μm
# B_λ(T) = 2hc² / λ⁵ / (exp(hc / λkT) − 1)
planck(2.2, 300.0) / L_reflected      # → 2.0 × 10⁻⁴           user
# τ_R(λ) ≈ 0.008569 λ⁻⁴ (1 + 0.0113 λ⁻² + 0.00013 λ⁻⁴)
tau_rayleigh(1.0) / tau_rayleigh(2.0) # → 16.1                 user
tau_rayleigh(2.2)                     # → 3.7 × 10⁻⁴           user

The band model and its linearisation¶

TL;DR. Average over the band, split the column into background and plume, and see how far a straight line carries.

Problem. Band $b$ integrates over its spectral response $f_b(\lambda)$. With the albedo flat across the band, $R_b = A_b\, \mathcal{T}_b(\Omega)$, where $\mathcal{T}_b$ is the band average of $e^{-m \sigma_\lambda \Omega}$, the exact band model of radtrax · 1. Split $\Omega = \Omega_{\text{bg}} + \Delta\Omega$, with $\Omega_{\text{bg}} = 0.0109$ kg/m² (1.9 ppm). The exact plume optical depth of band $b$ is $\tau_b(\Delta\Omega) = -\ln[\mathcal{T}_b(\Omega_{\text{bg}} + \Delta\Omega) / \mathcal{T}_b(\Omega_{\text{bg}})]$, and its tangent at the background is the sensitivity

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# Shapes: G = 100,000 wavelengths (0.01 nm); m = 2.225
srf_s2 = read_srf("S2A")         # 1 nm table, one column per band  today
# f_b(λ) on the 0.01 nm grid, zero outside the table          (G,) each
f11 = np.interp(WL, srf_s2.index, srf_s2["B11"], left=0, right=0)  # user
f12 = np.interp(WL, srf_s2.index, srf_s2["B12"], left=0, right=0)  # user
# σ̄_b = ⟨σ⟩_b: 0.040 and 0.400 m²/kg (chosen)
# α_b = −(1/m) ∂ ln T_b / ∂Ω at Ω_bg       → 0.0398, 0.3953  user
# m α₁₂ ΔΩ / τ₁₂(ΔΩ) at ΔΩ = 0.3 kg/m²     → 1.16           user
# nB_b(ΔX) = ⟨exp(−m σ ΔX)⟩_b, no background         (181,)
nb = build_nb_lut(ds, srf, T_K=280, p_atm=0.8, amf=m)       # today
# −ln nB₁₂ / τ₁₂ at 0.3 kg/m²                → 1.011          user

$\alpha_b$ sits just below $\bar\sigma_b$, because the background has already darkened the strongest line cores. At 0.3 kg/m² the straight line is 16 % above the exact optical depth, the curve of growth. Band 11, the 1.66 μm band’s edge, absorbs a tenth as much as band 12 on this spectrum (a modelling choice; it is not zero on real spectra either). plumax’s band table, nB, integrates the plume without the background column, which overstates band 12’s plume optical depth by 1.1 %.

Three ways to remove the surface¶

TL;DR. Ratio across passes (SBMP), across bands (MBSP), or both (MBMP); each ratio cancels what it assumes is constant.

Problem. Write (4) for bands 11 and 12, on a plume pass $(p)$ and a plume-free reference pass $(r)$ over the same pixels. The three estimators are:

Each scale factor is a least-squares fit through the origin over background pixels: $c_s$ from $R_{12}^{(r)} \approx c_s R_{12}^{(p)}$, and $c$ per pass from $R_{11} \approx c\, R_{12}$. The leftover log is what each method assumes away. SBMP needs the band-12 albedo unchanged between passes, up to one scene-wide factor. MBSP needs the band ratio $r$ the same everywhere in the scene, up to $c$. MBMP needs only $r$ unchanged at each pixel between passes, up to one scene-wide factor per pass: it is a difference in differences, across bands and then across dates. The column comes from inverting the exact table, $\Delta\Omega = \tau_{12}^{-1}(s)$ for SBMP and $\Delta\Omega = g^{-1}(s)$ with $g = \tau_{12} - \tau_{11}$ for MBSP and MBMP, or, to first order, $s / (m\, \alpha_{12})$ and $s / (m\, \Delta\alpha)$ with $\Delta\alpha = \alpha_{12} - \alpha_{11}$.

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# Shapes: obs, ref (2, 200, 200) = (B11, B12) × (H, W)
# c = Σ x y / Σ x² over background pixels                 () per fit
c_s = fit_c(R12r, R12p, bg)                                # user
c_p, c_r = fit_c(R11p, R12p, bg), fit_c(R11r, R12r, bg)    # user
# δ = c R₁₂ / R₁₁ − 1, per pass               (200, 200) → (200, 200)
d_p, d_r = c_p * R12p / R11p - 1, c_r * R12r / R11r - 1    # user
# ΔΩ = τ₁₂⁻¹(−ln(c_s R₁₂ / R₁₂′))              (200, 200) → (200, 200)
sbmp = invert(-np.log(c_s * R12p / R12r), tau12_tab)       # user
# ΔΩ = g⁻¹(−ln(1 + δ))                          (200, 200) → (200, 200)
mbsp = invert(-np.log1p(d_p), g_mb)                        # user
# ΔΩ = g⁻¹(−ln((1 + δ) / (1 + δ′)))             (200, 200) → (200, 200)
mbmp = invert(-(np.log1p(d_p) - np.log1p(d_r)), g_mb)      # user
# ln(R₁₁/R₁₂) − ln(R₁₁′/R₁₂′), no scale factors   (2, 200, 200) → ()
gz = gz.plume.SBMP(swir1=0, swir2=1, axis=0)(obs, ref)     # today

The running example: one scene, every nuisance¶

TL;DR. One plume, two passes, and every surface and atmospheric change the derivations warn about, each in its own patch.

Problem. A 200 × 200 scene with 20 m pixels holds a plume of peak 0.3 kg/m² (148,700 kg) on the plume pass only. Both passes share a textured surface: albedo varies by 15 %, the band ratio $r$ by 4 %, and a mineral patch has $r$ 15 % low. Between the passes the whole scene brightens 5 %, a field is wetted (both bands × 0.8), one patch’s ratio drops 3 %, and band 12’s water-vapour transmission drops 1 % everywhere. Pixels enter through plumax’s inject_plume, with the band table built against the background column.

The hierarchy is as the derivations say (Figure 1, Table 1). SBMP, with or without a scale factor, reads the wetted field as 0.28–0.29 kg/m² of methane, nearly the plume’s peak; MBSP reads the stable mineral patch as 0.21 kg/m² and the 4 % ratio texture as 0.054 kg/m² of clutter; MBMP removes both, and the scene-wide brightening and water-vapour change with them. What it keeps is the patch whose band ratio changed: 3 % reads as 0.039 kg/m². The plume mass then depends on two choices the derivations make differently. A scale factor fitted over every pixel takes in the plume, moves by 0.31 % and leaves a −0.009 kg/m² offset that costs 11 % of the mass. And the slope that converts the signal to a column matters: the exact table recovers 99 %, a linear inversion 91 %, the $\delta$ difference 88 %, and $\bar\sigma_{12}$ with $\sigma_{11} = 0$ 78 %. geotoolz’s SBMP keeps the 1 % water-vapour change as a +0.013 kg/m² offset over the whole scene, 16 % extra mass on this plume (why).

Noise¶

TL;DR. Each ratio cancels a surface term and adds the noise of the bands it uses.

Problem. With independent noise of relative size $\varepsilon$ in each band and pass, the log ratios add variances: SBMP uses two band-12 images, MBSP one image of each band, MBMP four images, and the two-band methods divide by the smaller $\Delta\alpha$. To first order,

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# Shapes: obs, ref (2, 200, 200); ε = 0.01 per band and pass
# sd(ΔΩ̂_noisy − ΔΩ̂_clean) off the plume       (200, 200) → ()
# → SBMP 0.016, MBSP 0.018, MBMP 0.025 kg/m²; theory within 3 %

As (6) predicts, MBMP pays for its robustness with 1.6 times SBMP’s noise. On a surface that is stable in time, SBMP with a scale factor is the more precise retrieval; on a surface that is uniform in its band ratio, MBSP needs no reference pass. MBMP is the default because real surfaces are neither, and its noise falls with averaging where a surface artefact does not.

When MBMP is not the right choice¶

Derivation (1) lists three cases, and the run bears on each:

1. No good reference pass. MBMP and SBMP both need a plume-free date with the same surface. If the only reference is from another season, the local ratio changes grow and MBSP, which needs none, may do better.

2. A strongly wavelength-dependent surface. Where $r$ varies widely across the scene, one $c$ fits it poorly; MBMP still cancels the stable part of $r$ (the mineral patch is 0.000 kg/m² above the offset), but the fitted $c$ is less well defined, and SBMP avoids the band ratio.

3. Water vapour that changes between passes. Band 12 has water-vapour lines. A scene-wide change is a scene-wide factor and the per-pass $c$ absorbs it; a change that varies across the scene (a front, terrain) does not, and reads like a changed band ratio: 1 % is 0.013 kg/m².

What each method cancels¶

Table 2 collects the run: what each method removes, what it leaves, and its noise.

Where it lives in the stack¶

The package boundaries inventory lists the pieces: plumax’s radtran.nb_lut (build_nb_lut, lookup_nb, inject_plume) for the band tables, georeader’s read_srf for the real response functions (B11 1569–1658 nm, B12 2113–2286 nm at half maximum), and geotoolz’s plume.SBMP for the two-pass, two-band log signal. The scale factors, the background masks and the exact inversion exist nowhere in the stack; the notebook writes them.

Open questions¶

1. What should the operators be called? An MBMP operator in radtrax, with SBMP and MBSP beside it, each with the scale factors and a background mask, would match the literature; the current SBMP could become MBMP’s scale_factors=False mode.

2. Where does the background mask come from? The run uses a first-pass detection grown by a Gaussian of 2 pixels; an iterated fit, or the plume mask from geotoolz’s PlumeMask, are the alternatives, and the choice moves the offset.

3. Should the band table include the background? plumax’s nB omits it; including it is one extra integral and changes band 12 by 1 % here, more over high background or long air-mass paths.

Upcoming work¶

Table 3 lists the follow-ups.

References¶

• The two derivations: “Deriving the MBMP method family from first principles” Johnson, 2026 and the step-by-step optical derivation Johnson, 2026.

• The SBMP, MBSP and MBMP retrievals Varon et al., 2021; the IME and the fine-scale plume retrievals they feed Varon et al., 2018; a ground-truth calibration of Sentinel-2 retrievals Chen & others, 2022; transient emissions from geostationary observations Watine-Guiu et al., 2023.

• geotoolz, georeader and plumax at the revisions in the header.