papers/zoo-carbon-credits/sections/03-carbonnet-biomass-estimation-model.tex

\section{CarbonNet: Biomass Estimation Model}

\label{sec:carbonnet}
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\subsection{Architecture}

CarbonNet is a multi-modal deep learning model that fuses three data
streams to estimate above-ground biomass:

\begin{enumerate}
  \item \textbf{Optical branch}: A ResNet-50 encoder processes
    Sentinel-2 Level-2A surface reflectance imagery (13 spectral bands,
    10--60\,m resolution) through spatial and spectral attention.
  \item \textbf{Radar branch}: A separate ResNet-50 encoder processes
    Sentinel-1 SAR backscatter (VV and VH polarizations, 10\,m
    resolution), which penetrates cloud cover and is sensitive to
    woody biomass.
  \item \textbf{LiDAR branch}: A point cloud encoder processes GEDI
    full-waveform LiDAR returns to extract canopy height profiles.
\end{enumerate}

The three branches are fused through a cross-attention mechanism:
\begin{equation}
  \mathbf{z} = \text{CrossAttention}(\mathbf{h}_{\text{opt}},
  \mathbf{h}_{\text{sar}}, \mathbf{h}_{\text{lidar}}) \in \R^{256}
\end{equation}

A regression head maps $\mathbf{z}$ to AGB in Mg/ha:
\begin{equation}
  \hat{B} = \text{MLP}(\mathbf{z}) \in \R_{\geq 0}
\end{equation}

\subsection{Training Data}

\paragraph{GEDI-Biomass training set.} We use 4.2 million GEDI
footprints (25\,m diameter) with associated AGB estimates from the GEDI
Level-4A product \citep{dubayah2022gedi}. Each footprint is paired with
co-located Sentinel-2 and Sentinel-1 imagery within a 30-day window.

\paragraph{Field plot calibration.} We further calibrate against 12{,}847
field inventory plots from the Forest Observation System
\citep{schepaschenko2019forest}, spanning tropical, temperate, and boreal
forests across 47 countries.

\subsection{Loss Function}

We minimize a composite loss:
\begin{equation}
  \Loss = \Loss_{\text{MSE}} + \lambda_1 \Loss_{\text{spatial}} +
  \lambda_2 \Loss_{\text{uncertainty}}
\end{equation}

where $\Loss_{\text{MSE}}$ is the mean squared error on AGB,
$\Loss_{\text{spatial}}$ is a spatial smoothness regularizer penalizing
discontinuities at tile boundaries, and $\Loss_{\text{uncertainty}}$ is
the negative log-likelihood of a heteroscedastic Gaussian that
encourages well-calibrated uncertainty estimates.

\subsection{Results}

\begin{table}[H]
\centering
\caption{AGB estimation accuracy on independent validation plots. RMSE
in Mg/ha. Best results in \textbf{bold}.}
\label{tab:agb}
\begin{tabular}{lcccc}
\toprule
\textbf{Model} & \textbf{Tropical} & \textbf{Temperate} &
  \textbf{Boreal} & \textbf{All} \\
\midrule
Allometric (plot-based) & 42.1 & 31.7 & 24.3 & 35.2 \\
Random Forest & 31.4 & 24.8 & 19.1 & 27.6 \\
GBM (XGBoost) & 29.8 & 23.1 & 18.4 & 25.9 \\
ESA CCI Biomass & 33.7 & 25.6 & 20.7 & 28.3 \\
Lang et al.\ (2023) & 26.2 & 20.4 & 16.8 & 22.1 \\
\textbf{CarbonNet} & \textbf{22.7} & \textbf{17.3} &
  \textbf{13.8} & \textbf{18.4} \\
\bottomrule
\end{tabular}
\end{table}

CarbonNet achieves 18.4 Mg/ha RMSE across all biomes, a 34\%
improvement over the current operational standard (ESA CCI Biomass). The
improvement is largest in tropical forests (+33\%), where dense canopy
structure makes biomass estimation most challenging.


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