The forward problem
The forward model maps three unknown parameters, θ = (τ₁, τ₂, f), to a predicted observation M(t): the sum of two exponentially decaying components. In this application, the components represent stores of decomposing plant material, called pools in carbon-cycle science. Each loses material at a rate proportional to its remaining amount, and observations measure their combined remaining mass at specified times.
Component 1: f
decay time τ₁
+Component 2: 1 − f
decay time τ₂
dC₁/dt = −C₁/τ₁
dC₂/dt = −C₂/τ₂
M(t) = 100[f exp(−t/τ₁) + (1−f) exp(−t/τ₂)]
The inverse problem estimates θ from the observations yᵢ at times tᵢ, accounting for assumed errors, fixed parameter bounds and the chosen coordinates. The posterior describes the parameter combinations supported by those choices and the data. Here, the observed quantity is the percentage of plant organic matter remaining; this example illustrates carbon-pool inference using a simplified model.
The observations used here
| Time (months) | Organic matter remaining (%) |
|---|
| 0 | 100.00 · initial normalization |
| 2.5 | 84.21 |
| 6 | 73.09 |
| 12 | 61.08 |
| 24 | 47.23 |
The observations are Bouteloua gracilis organic matter remaining at Cedar Creek under the UV-block treatment, from Brandt et al. (2010), archived in aridec v1.0.2. The numerical series is column CBB of the archived Brandt2010 entry. The time-zero normalization is fixed by the model and is excluded from the likelihood. Only the chosen observations enter each fit.
Original field experiment: Brandt et al. (2010)
Archived database: aridec v1.0.2
Exact numerical source · Series metadata
Why this solution space is interesting
Two modes can describe exactly the same decay curve. Swapping τ₁ and τ₂ while replacing f by 1−f leaves M(t) unchanged. Those two peaks represent the same model prediction under different component labels. “Swap component names” demonstrates this with the same predicted observations.
The data also leave genuine parameter trade-offs. Different component fractions and decay times can produce similar predictions at a few observation times. Use the first two observations to expose broad curved regions, then restore all four. At the default error scale, the region where the two decay times are equal is much less plausible with all four observations. Any visible connection depends on the fixed display threshold and the finite grid.
A local quadratic approximation cannot explain the whole geometry. The component-label symmetry and curved trade-offs arise from the forward equations and these observations. No extra peaks, bridges, or warping functions were added to the posterior.
Statistical assumptions, coordinates, and numerical checks
The likelihood uses independent Gaussian residuals, with the selected common σ. The prior is uniform in (log₁₀τ₁, log₁₀τ₂, f), with decay times between 10−0.3 and 102.3 months (about 0.50–200 months), and f between 0.02 and 0.98. Thus the density shown is with respect to log decay-time coordinates and the fraction, not linear decay-time coordinates. These prior bounds and the error model are analysis choices, not claims from the original study.
J(θ) = Σᵢ [M(tᵢ; θ) − yᵢ]² / (2σ²)
p(θ | y) / pmax = exp[−(J(θ) − Jmin)]
p(log τ₁, log τ₂ | y) ∝ ∫ p(log τ₁, log τ₂, f | y) df
The blue–turquoise–yellow color scale is applied to log relative posterior density. Transparency helps expose the volume interior, so overlapping dots can blend colors; use the 2D map to avoid overlapping dots.
The three-parameter cloud uses a regular 65 × 65 × 81 grid in the displayed prior coordinates. Its dots are volume-rendering elements, not Monte Carlo samples. The flat two-parameter map marginalizes over f by midpoint quadrature (121 nodes). Color shows relative log marginal density. The two views each have their own maximum for the color scale. Peak density and integrated probability mass are different quantities.
For the selected observations, the optimum fraction at any pair of decay times is found analytically by bounded linear least squares. A grid search followed by a pattern search locates a best fit. For all four observations, an independent numerical optimization gives approximately τfast = 3.284 months, τslow = 46.903 months, ffast = 0.2139, with squared residual sum 0.4475 percentage-points². The model and observations are embedded in this file; no server or network connection is needed. Three.js is included, with a Canvas fallback when WebGL is unavailable.
With only four nonzero-time observations and a simplified stationary model, this is an explainer of an actual inverse problem, not a validated ecological parameter product. It does not model seasonal forcing, microbial transfers, or every cause of litter loss. Broader discussion of identifiability in decomposition data: Sarquis & Sierra (2023).