Parameters:
Northern flux · Southern flux · Mixing time
Posterior density
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What do these parameters mean?
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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.
Northern flux · Southern flux · Mixing time
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.
Choose values for the unknown quantities, such as a decay time or an emission rate.
Run the physical model. Compare its prediction with the measurements, allowing for errors.
Examine all supported combinations. Ask which new observation would tell them apart.
| Example | What we observe | What we infer |
|---|---|---|
| Land carbon | Plant material remaining over time | Decay times and starting amounts |
| Atmospheric remote sensing | Light absorbed at different wavelengths | Gas amount, light path, surface brightness |
| Atmospheric fluxes | Gas concentrations and isotope ratios | Sources, atmospheric mixing, or chemical loss |
| Seismology (analogy) | Seismic-wave arrival times | Earthquake 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.
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.
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.
Each box contains half the atmospheric air mass. Surface fluxes enter their respective hemispheres, and exchange moves gas between the two boxes. Exchange conserves the global burden.
The definition of τ follows the original Python model: without sources or loss, the north–south difference decays as exp(−t/τ). The exchange coefficient in each box is 1/(2τ).
CO₂ mode: λ = 0. Fₙ and Fₛ are net surface fluxes, combining emissions and land/ocean exchange; positive flux adds carbon to the atmosphere. They are constant over this experiment. The plotted CO₂ anomaly starts at zero in both boxes relative to a fixed initial state. This is a prescribed net-flux experiment, without a separate land/ocean reservoir or an OH sink.
We use v = 1/1.062 ppm per PgC for a single hemisphere, corresponding to 2.124 PgC per global ppm. Thus the global CO₂ increase depends on Fₙ + Fₛ. Hemispheric differences add information about where carbon enters and how rapidly it mixes.
CH₄ mode: the unknowns are global biogenic emissions Eᵦ, global fossil emissions E𝒇, and a shared OH loss factor h. Northern source fractions are fixed at 0.65 and 0.90, respectively. The transport time is fixed at 1 year, and the reference methane lifetime is 11.5 years.
The normalized isotope-abundance tracers Q₁₃ and Qᴰ are transported like bulk methane, but their chemical loss is slower. This uses the dilute-isotope approximation; isotope ratios are recovered after transporting and removing abundance tracers. A source supplies Q at its emission rate multiplied by (1 + δsource/1000).
| Toy source / sink | δ¹³C (‰) / KIE | δD (‰) / KIE |
|---|---|---|
| Biogenic source | −62 | −320 |
| Fossil source | −44 | −192 |
| Kinetic isotope effect | 1.0065 | 1.276 |
The two source fingerprints are fixed illustrative endmembers. KIE values are dimensionless. The reference lifetime, KIEs, exchange convention, and methane conversion v = 0.7178 ppb per hemispheric Tg CH₄ follow the earlier Python box model. Here all methane loss scales with h; stratospheric loss, soil uptake, and chlorine are not separately resolved.
Dasgupta, Pandey et al. (2026): Global methane emission estimates from a dual-isotope inversion: new constraints from δD-CH₄, Atmospheric Chemistry and Physics, 26, 8601–8616.
The paper combines hemispheric methane and both isotope records in a Bayesian two-box inversion. Its full model has five source categories, multiple sink processes, and time-varying inferred emissions. Here the CH₄ → +δ¹³C → +δD sequence illustrates that measurement principle with two constant source totals and one loss multiplier.
This page uses a compact analytic implementation of the source, exchange, and first-order chemistry structure in Sudhanshu Pandey's twoboxCH4 code. All plotted observations are newly generated synthetic data. The page does not reproduce the paper's fitted time series, posterior estimates, or complete inversion method. The CO₂ option is a separate adaptation of the atmospheric exchange equations.
CO₂: the global mean constrains the total flux but contains no information about its hemispheric split or transport time. Two hemispheric measurements at one time constrain the total and a flux-difference–transport combination, producing a curved ridge. Measurements during the transient evolution help distinguish transport from the flux split.
CH₄: more emissions and stronger OH loss can compensate in the concentration record. Carbon isotopes add information about the source mixture. Hydrogen isotopes add a different combination of source and sink sensitivity. The amount of improvement depends on the source fingerprints, kinetic isotope effects, errors, and observation times assumed here.
“Another explanation” selects a second solution that matches the selected endpoint concentrations. Compare its other predictions to see what additional observations can distinguish. These trade-offs arise from the forward equations. No artificial modes or bridges are added.
CO₂ truth: (Fₙ, Fₛ, τ) = (5, 0, 1), with fluxes in PgC yr⁻¹ and τ in years. Observations at 0.25, 0.5, 1, 2, and 3 years have a base error of 0.16 ppm. The global-mean series is the average of the same paired hemispheric observations, with error 0.16/√2 ppm. It is used separately from those observations, so the likelihood never counts a derived mean and its constituents as independent data.
CH₄ truth: (Eᵦ, E𝒇, h) = (370, 130, 1), with emissions in Tg CH₄ yr⁻¹. Initial methane is 1900 ppb north and 1800 ppb south; δ¹³C is −47 and −47.3‰, and δD is −90 and −92‰. Observations at 0.5, 1, 2, 3, and 5 years have base errors of 6 ppb, 0.18‰ and 1‰. The initial state is known and excluded from the likelihood. The fixed Gaussian noise realizations use seeds 82 (CO₂) and 317 (CH₄).
Independent uniform priors in the linear display coordinates: CO₂ Fₙ ∈ [1, 8], Fₛ ∈ [−3, 3], τ ∈ [0.3, 2.5]; CH₄ Eᵦ ∈ [250, 550], E𝒇 ∈ [50, 220], h ∈ [0.6, 1.5]. These wide bounds and errors are teaching choices. Increasing the error multiplier changes the likelihood uncertainty without changing the observations.
For constant coefficients, mean and difference equations have exact exponential solutions. The page evaluates these analytic solutions directly, then computes the posterior on an 81 × 81 × 81 grid. A grid search followed by local refinement gives a representative best fit. Grid dots are not MCMC samples; the posterior is not a Hessian approximation.
The 2D map integrates out the third parameter using trapezoidal quadrature. The uncertainty table uses the same quadrature over the full 3D grid, including densities below the display threshold. Intervals are approximate grid quantiles. Each view is normalized to its own density maximum. The color mapping is logarithmic; 100% is peak density, not certainty.
Naus, Montzka, Pandey et al. (2019), Constraints and biases in a tropospheric two-box model of OH: discusses biases from representing transport, OH exposure, and observations with two hemispheric boxes. This explorer cannot resolve regional fluxes, vertical gradients, spatially varying chemistry, or station sampling.
Global Carbon Budget 2024, Table 1: supplies the global conversion 1 ppm CO₂ = 2.124 PgC. Using half that atmospheric mass per box is an idealized equal-hemisphere approximation.
| Time (yr) | Quantity | Hemisphere | Value | Assumed σ | Used |
|---|