Pilar 3 — 4D Pedometry: Stacked ConvLSTM Forecasts of SOC

Hugo Rodrigues

Abstract

Most digital soil maps report a time-invariant property field, which ignores the evidence that topsoil Soil Organic Carbon (SOC) responds measurably to climate forcing on monthly to annual scales (Lehmann and Kleber 2015; Minasny et al. 2017). The Pillar 3 of edaphos addresses that gap with a stacked Convolutional LSTM (Shi et al. 2015) trained in sequence-to-sequence mode plus a multi-step rollout wrapper for forward forecasting under known future drivers. A physics-informed mass-balance regulariser optionally penalises violations of an analytical SOC kinetic, fusing Pillar 2 into Pillar 3 (Raissi, Perdikaris, and Karniadakis 2019; Reichstein et al. 2019).

1. From 3D to 4D pedometry

Let $y(\mathbf{s},t)$ denote a topsoil property at location $\mathbf{s}$ and time $t\in\{1,\ldots,T\}$. Traditional DSM (McBratney, Mendonça Santos, and Minasny 2003) estimates $y(\mathbf{s})=\bar y(\mathbf{s},\cdot)$ by collapsing over time. A 4D model retains the time dimension and predicts the full spatio-temporal field $y(\mathbf{s},t)\mid \mathbf{X}(\mathbf{s},t^{\prime}\le t)$, where $\mathbf{X}$ is the driver stack (climate, vegetation, static topography).

The Convolutional LSTM cell (Shi et al. 2015) operationalises spatial memory: at each time step the hidden state $\mathbf{H}_{t}$ and the cell state $\mathbf{C}_{t}$ are tensors of the same spatial size as the input, so memory propagates with its location: $$\begin{aligned} \mathbf{i}_{t}&=\sigma(W_{xi}*\mathbf{X}_{t}+W_{hi}*\mathbf{H}_{t-1}+b_{i}),\\ \mathbf{f}_{t}&=\sigma(W_{xf}*\mathbf{X}_{t}+W_{hf}*\mathbf{H}_{t-1}+b_{f}),\\ \mathbf{g}_{t}&=\tanh(W_{xg}*\mathbf{X}_{t}+W_{hg}*\mathbf{H}_{t-1}+b_{g}),\\ \mathbf{o}_{t}&=\sigma(W_{xo}*\mathbf{X}_{t}+W_{ho}*\mathbf{H}_{t-1}+b_{o}),\\ \mathbf{C}_{t}&=\mathbf{f}_{t}\odot\mathbf{C}_{t-1}+\mathbf{i}_{t}\odot\mathbf{g}_{t},\\ \mathbf{H}_{t}&=\mathbf{o}_{t}\odot\tanh(\mathbf{C}_{t}), \end{aligned} \tag{1}$$ with * a 2-D convolution and $\odot$ the Hadamard product. Stacking $L$ cells feeds $\mathbf{H}_{t}^{(\ell)}$ as the input to layer $\ell+1$, yielding a hierarchy of spatial receptive fields.

2. Synthetic SOC dynamics cube

To keep the vignette self-contained and reproducible, [temporal_synth_soc_cube()][temporal_synth_soc_cube] integrates the driver-response kinetic $$\mathrm{SOC}_{t+1}=\mathrm{SOC}_{t}+k_{\text{in}}P_{t} -k_{\text{out}}\,\mathrm{SOC}_{t}\,P_{t}/\bar P +\varepsilon, \tag{2}$$ with $P_{t}$ the monthly precipitation field and $\bar P$ its long-term mean. The numerator $k_{\text{in}}P_{t}$ models organic input proportional to wet-season biomass turnover; the denominator $k_{\text{out}}\mathrm{SOC}\,P/\bar P$ captures humidity-modulated decomposition (Lehmann and Kleber 2015; Minasny et al. 2017).

library(edaphos)
.torch_ok <- requireNamespace("torch", quietly = TRUE) &&
             isTRUE(tryCatch(torch::torch_is_installed(),
                             error = function(e) FALSE))
if (!.torch_ok) {
  knitr::knit_exit(
    "torch runtime (libtorch) not available — skipping vignette."
  )
}
cube <- temporal_synth_soc_cube(H = 12L, W = 12L, T_total = 18L,
                                seed = 7L)
str(cube)
#> List of 3
#>  $ elev  : num [1:12, 1:12] 42.8 32.6 42.4 51.1 55.6 ...
#>  $ precip: num [1:18, 1:12, 1:12] 16.3 28.7 53.2 71.1 87.7 ...
#>  $ soc   : num [1:18, 1:12, 1:12] 18.9 19.4 20.1 21.3 23.3 ...

torch runtime (libtorch) not available — skipping vignette.

Lehmann, J., and M. Kleber. 2015. “The Contentious Nature of Soil Organic Matter.” Nature 528: 60–68. https://doi.org/10.1038/nature16069.

McBratney, A. B., M. L. Mendonça Santos, and B. Minasny. 2003. “On Digital Soil Mapping.” Geoderma 117 (1-2): 3–52. https://doi.org/10.1016/S0016-7061(03)00223-4.

Minasny, B., B. P. Malone, A. B. McBratney, D. A. Angers, D. Arrouays, A. Chambers, V. Chaplot, et al. 2017. “Soil Carbon 4 Per Mille.” Geoderma 292: 59–86. https://doi.org/10.1016/j.geoderma.2017.01.002.

Raissi, M., P. Perdikaris, and G. E. Karniadakis. 2019. “Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations.” Journal of Computational Physics 378: 686–707. https://doi.org/10.1016/j.jcp.2018.10.045.

Reichstein, M., G. Camps-Valls, B. Stevens, M. Jung, J. Denzler, N. Carvalhais, and Prabhat. 2019. “Deep Learning and Process Understanding for Data-Driven Earth System Science.” Nature 566: 195–204. https://doi.org/10.1038/s41586-019-0912-1.

Shi, X., Z. Chen, H. Wang, D.-Y. Yeung, W.-K. Wong, and W.-C. Woo. 2015. “Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting.” In Advances in Neural Information Processing Systems, 28:802–10.