Inverse Modeling Explorer

We “invert” measurements to infer processes such as carbon storage or greenhouse-gas emissions. A good fit can have several explanations: explore which answers the data support and which new measurements help tell them apart.

Parameters:

Methane amount · Light path · Surface brightness

Posterior density
LowHigh
X q · methane (1 = 1900 ppb)Y m · light path (1 = reference)Z A · reflected fraction
Preparing the explorer… JavaScript is needed for the interactive plots.
What do these parameters mean?

Methane absorbs some wavelengths of sunlight. A longer path through the gas causes more absorption; a brighter surface reflects more light toward the instrument.

q · Methane amount
Scale relative to 1900 parts per billion (ppb): q = 1.2 means 2280 ppb for the assumed air column.
m · Light-path factor
Scale relative to a reference light path. A larger value means light encounters more gas along its path.
A · Surface brightness (albedo)
Fraction of incoming light reflected by the surface. A = 0.3 means 30% is reflected. We assume the same albedo in both gas bands.

Try this:

The same reasoning across Earth science

A forward model predicts observations from a set of parameters. An inverse problem uses observations to work back to plausible parameters. The difficulty is that different parameters can produce similar signals.

1 · Propose an explanation

Choose values for the unknown quantities, such as a decay time or an emission rate.

2 · Predict and compare

Run the physical model. Compare its prediction with the measurements, allowing for errors.

3 · Keep the uncertainty

Examine all supported combinations. Ask which new observation would tell them apart.

ExampleWhat we observeWhat we infer
Land carbonPlant material remaining over timeDecay times and starting amounts
Atmospheric remote sensingLight absorbed at different wavelengthsGas amount, light path, surface brightness
Atmospheric fluxesGas concentrations and isotope ratiosSources, atmospheric mixing, or chemical loss
Seismology (analogy)Seismic-wave arrival timesEarthquake location and wave speeds

Why the shape matters: a long band means quantities can compensate for each other. Separate regions can mean different explanations, or simply different names for the same components. A compact region means the parameters are better constrained under the chosen model and assumptions.

More measurements help most when they respond differently to the unknowns. Even a very good fit can give uncertain predictions, especially beyond the times or conditions observed.

The Bayesian vocabulary, in plain language
Parameter
An unknown quantity we want to estimate.
Prior
The ranges and relative plausibility assigned before using these observations.
Likelihood
How well a prediction explains the observations under an assumed error model.
Posterior
The probability distribution over parameters after combining the likelihood and prior.
Marginalization
Adding up posterior density over a parameter to describe the others. The 2D map does this for the third parameter.

posterior ∝ likelihood × prior

Here, the prior is constant in the linear parameter coordinates inside the stated bounds and zero outside. Those bounds can still influence the uncertainty.

A best fit is a single combination. The posterior also describes uncertainty around it. Probability belongs to a region of parameter space: the color of a single point is a density, expressed here relative to the peak.

Equations, assumptions, data and references

Prior choice: No external estimates of the three fitted quantities are used as priors. Within the chosen ranges, the prior is uniform in the plotted coordinates, so the 3D density is proportional to the data likelihood.

The display omits density below exp(−6), about 0.25% of the peak, for readability. The calculated uncertainty includes the full bounded posterior. Each selection of observations and each view has its own peak, so compare the width and shape of the supported region when changing observations. The cloud shows grid locations, with transparent dots that can overlap. The 2D map integrates over the third parameter; selecting a map location shows its conditional best-fitting third value, which is a representative prediction for that location. All three examples use a teaching model, chosen error assumptions, and finite parameter ranges. The uncertainty shown covers only the three fitted quantities; fixed inputs and model structure are not varied.

A small radiative-transfer inverse problem

For a reflecting surface under an absorbing atmosphere, normalized reflected radiance is the surface albedo multiplied by the two-way transmission. We combine the known solar and viewing geometry with the reference gas optical depths, and write:

Rⱼ(q, m, A) = A exp[−m(q kCH₄,ⱼ + kCO₂,ⱼ)]
yⱼ = Rⱼ(1.20, 1.05, 0.32) + εⱼ
J(θ) = Σⱼ∈used [Rⱼ(θ) − yⱼ]² / (2σ²)
p(θ | y) ∝ exp[−J(θ)] inside the prior bounds

R is πL divided by incident solar irradiance and the cosine of solar zenith angle. The coefficients k are dimensionless reference optical depths: gas cross section × reference column × reference air mass factor. CH₄ channels have kCO₂ = 0; CO₂ channels have kCH₄ = 0. The CO₂ column is treated as known.

All observations and optical-depth coefficients on this page are synthetic. They illustrate absorption and identifiability. Channel numbers are schematic spectral samples, not calibrated wavelengths or HITRAN lines. The chosen methane reference of 1900 ppb is a scaling convention.

What each extra measurement reveals

ObservationsConstrained combinationsGeometry
One CH₄ channelA exp(−kqm)Curved sheet: two unconstrained local directions
Twelve CH₄ channelsA and q × mCurved ridge: one unconstrained local direction
CH₄ + known CO₂A, q and mLocalized region: three locally identifiable directions

The CH₄-only spectrum is exactly unchanged by q → cq and m → m/c within the bounds. More CH₄ channels constrain their product, but cannot separate its factors in this model. CO₂ absorption responds to m independently of q, which breaks this ambiguity.

A ridge is a continuous family of solutions. This model does not require separated modes. Its non-Gaussian shape comes from the exponential forward model and the product q × m.

References and connection to operational retrievals

Jongaramrungruang et al. (2021), Remote sensing of methane plumes: instrument tradeoff analysis for detecting and quantifying local sources at global scale, AMT 14, 7999–8017: equations 1–2 supply the reflected-sunlight and Beer–Lambert structure used here. That study includes realistic spectra, surface reflectance and instrument effects; this page reduces the physics to a few idealized channels.

Chan Miller et al. (2024), Methane retrieval from MethaneAIR using the CO₂ proxy approach, AMT 17, 5429–5454: motivates using CO₂ to constrain the common absorption path. This toy jointly fits a single effective path factor; it is a pedagogical analogue of the proxy principle, not a reproduction of their retrieval algorithm.

The simplifications are a fixed atmospheric state, a known CO₂ column, a common wavelength-independent path factor, and one grey albedo for both bands. There is no explicit multiple scattering, vertical profile, interfering water vapor, instrument convolution, wavelength shift, or uncertain CO₂. Real instruments need these effects treated or screened. In particular, scattering does not generally reduce to one scalar path factor.

Prior, uncertainty, and numerical method

Independent uniform priors: 0.5 ≤ q ≤ 2.5; 0.5 ≤ m ≤ 1.7; 0.16 ≤ A ≤ 0.48. Density is with respect to these linear coordinates. Likelihood errors are independent Gaussian with the selected σ. The fixed synthetic noise realization was generated with standard deviation 0.006 (seed 42); changing σ changes the assumed uncertainty only.

A regular 81 × 81 × 81 grid evaluates the 3D posterior. Dots are grid locations, not Monte Carlo samples. The 2D map integrates out albedo using 121 midpoint nodes and rescales to its own maximum. The uncertainty table uses tensor-product trapezoidal weights over the full 3D grid, including regions below the display threshold; central intervals are approximate grid quantiles.

At each (q, m), albedo is linear in radiance, so its best fit is solved analytically and clipped to the bounds. An 81 × 81 search followed by local refinement finds a representative best fit. With CH₄ alone, equally good fits exist along the ridge or sheet. A best-fit marker represents one of them.

No MCMC, Hessian approximation, or sampling benchmark is used on this page. The forward model and data are embedded, so this file runs offline after downloading.

Exact synthetic channels and observations

One-channel mode uses channel 4. CH₄ mode uses channels 1–12; the combined mode adds channels 13–20. Lines connecting channels in the spectrum are visual guides.

ChannelBandkCH₄kCO₂y (normalized radiance)Used