% ========================
% Appendix Section
% ========================

%%%%%%%%% APPENDIX
\clearpage
\appendix
\section*{Appendix}
\renewcommand{\thesection}{A}

\setlength{\abovecaptionskip}{4pt}
\setlength{\belowcaptionskip}{-4pt}
\setlength{\intextsep}{6pt}
% -------udaiveer-------------
% ========================
% Extended Implementation Details (SpectralMAE)
% ========================

\subsection{Pretraining Dataset}
\paragraph{Dynamic World Land Cover Classes.}
The pretraining and evaluation datasets for all modalities use land-cover annotations derived from the \textbf{Dynamic World (DW)} dataset, a global 10\,m resolution product from Google Earth Engine that provides near real-time land-cover classification from Sentinel-2 imagery.  
Dynamic World defines ten surface classes: \textit{(0) Water, (1) Trees, (2) Grass, (3) Flooded Vegetation, (4) Crops, (5) Shrub and Scrub, (6) Built Area, (7) Bare Ground} and \textit{(8) Snow and Ice}.  
Each pixel in our dataset is associated with one of these classes, ensuring consistent sampling and class-balanced training across modalities.  
Figure~\ref{fig:landcover_maps} illustrates the spatial distribution of Sentinel-2 and PlanetScope samples across India and neighbouring regions.

\vspace{7pt}
\begin{figure*}[h]
    \centering
    % --- Sentinel-2 Landcover ---
    \begin{subfigure}[t]{0.48\textwidth}
        \includegraphics[width=\linewidth, height=1.02\linewidth]{img/Sentinel_lancover.jpg}
        \caption*{(a) Sentinel-2 sampled pixels - visualization of 100,000 randomly sampled pixels per Dynamic World class across India and neighbouring regions, colored by class label. Data sourced from Google Earth Engine.}
        \label{fig:sentinel_landcover}
    \end{subfigure}
    \hfill
    % --- Planet Landcover ---
    \begin{subfigure}[t]{0.48\textwidth}
        \includegraphics[width=\linewidth, height=1.02\linewidth]{img/Planet_grid.jpg}
        \caption*{(b) PlanetScope sampled grids - each grid represents uniformly sampled Planet imagery covering identical geographic regions. Planet samples are spatially gridded rather than pixel-wise. Data sourced from Planet Explorer.}
        \label{fig:planet_landcover}
    \end{subfigure}

    \vspace{5pt} % space between images and caption
    \caption{\textbf{Dynamic World land-cover sampling across modalities.} 
    (a) Sentinel-2 pixel-wise sampling and (b) PlanetScope grid-based sampling across India and neighbouring countries. 
    These samples represent the nine Dynamic World land-cover classes and are used to ensure consistent cross-modal pretraining and evaluation.}
    \label{fig:landcover_maps}
\end{figure*}

\vspace{4pt}
\noindent\textit{Note:} These land-cover classes form the shared label space used for downstream multimodal fusion and evaluation, allowing for consistent comparison across optical, radar, and climate encoders before fusion in the results section.

\vspace{8pt}
\begin{table}[h]
\centering
\small
\setlength{\tabcolsep}{6pt}   % narrower columns to avoid text spill
\renewcommand{\arraystretch}{1.05}

\caption{\textbf{Spectral bands for Sentinel-2 and PlanetScope.}
Central wavelength and bandwidth (nm) used in all pretraining experiments.}
\label{tab:s2_ps_bands}

\begin{minipage}{0.48\textwidth}
\centering
\begin{tabular}{@{}l l c c@{}}
\toprule
\textbf{Sentinel-2 Band} & \textbf{Description} & \textbf{Center} & \textbf{BW} \\
\midrule
B2  & Blue          & 490  & 65  \\
B3  & Green         & 560  & 35  \\
B4  & Red           & 665  & 30  \\
B5  & Red Edge 1    & 705  & 15  \\
B6  & Red Edge 2    & 740  & 15  \\
B7  & Red Edge 3    & 783  & 20  \\
B8  & NIR           & 842  & 115 \\
B8A & NIR (narrow)  & 865  & 20  \\
B11 & SWIR 1        & 1610 & 90  \\
B12 & SWIR 2        & 2190 & 180 \\
\bottomrule
\end{tabular}
\end{minipage}

\vspace{6pt}

\begin{minipage}{0.48\textwidth}
\centering
\begin{tabular}{@{}l l c c@{}}
\toprule
\textbf{PlanetScope Band} & \textbf{Description} & \textbf{Center} & \textbf{BW} \\
\midrule
B1 & Coastal Blue  & 443 & 20 \\
B2 & Blue          & 490 & 50 \\
B3 & Green I       & 531 & 14 \\
B4 & Green II      & 565 & 31 \\
B5 & Red           & 665 & 31 \\
B6 & Red Edge      & 705 & 15 \\
B7 & NIR I         & 865 & 40 \\
B8 & NIR II        & 945 & 40 \\
\bottomrule
\end{tabular}
\end{minipage}
\end{table}


\subsection{Implementation Details}
\subsubsection{SpectralMAE for Multispectral Imagery}

The implementation follows a fully MAE-compliant setup with several practical details.  
Each training batch samples masked and visible bands through random noise ranking, ensuring stochastic masking patterns per iteration.  
Learnable encoder and decoder mask tokens, as well as the CLS token, are initialized using truncated normal noise ($\sigma{=}0.02$).  
All transformer modules operate under mixed-precision training (\texttt{torch.amp.autocast}) with gradient scaling, and matrix multiplication precision is explicitly set to \texttt{high} for numerical stability on GPUs.  
Masked reconstruction loss is applied only to hidden spectral positions using boolean masking, preventing gradients from unmasked inputs.  
Training batches of 2048 samples are loaded with pinned memory and four worker threads; all remaining samples are retained (\texttt{drop\_last=False}).  
Early stopping monitors validation loss with a patience of 15 epochs and a minimum improvement threshold of $10^{-6}$.  
The best checkpoint is reloaded and only the encoder weights are exported for downstream tasks.  
A complete training report is automatically generated, including CSV logs, a PDF summary with loss curves, and a t-SNE visualization of CLS embeddings (perplexity = 35).  
These implementation components ensure reproducible and hardware-efficient pretraining consistent with MAE principles.


\subsubsection{ClimateMAE for Environmental Variables}

Several low-level implementation features support stable and geographically consistent pretraining.  
Masked feature indices are resampled independently for every batch using uniform random scores, and masking is performed by direct zero-filling of the corresponding feature values (rather than replacing them with a learnable token).  
Both the encoder and decoder initialize learnable parameters, including the residual mask token and per-feature query embeddings, with truncated normal noise ($\sigma{=}0.02$).  
Latitude and longitude are passed in raw degrees to preserve periodic structure and are never standardized by the feature scaler.  
Mixed-precision training (\texttt{torch.amp.autocast}) with dynamic gradient scaling is enabled throughout, and matrix multiplication precision is set to \texttt{high} for reproducible GPU behavior.  
The geographic split employs integer block identifiers computed from $\lfloor\text{lat}/0.9^\circ\rfloor$ and $\lfloor\text{lon}/0.9^\circ\rfloor$ to ensure spatially disjoint training and validation sets; a diagnostic check reports mean and median nearest-neighbor distances (in km) to verify spatial independence.  
Training and validation histories are streamed to a CSV log, and the best checkpoint is retained based on validation MAE improvement exceeding $10^{-6}$.  
After convergence, the encoder weights and the full set of CLS embeddings are exported to disk for downstream use, enabling later retrieval without re-running pretraining.



\subsubsection{BYOL-Denoise for Sentinel-1 Radar}

The training pipeline follows a fully deterministic and mixed-precision implementation.  
Model parameters of the target (EMA) branch are initialized as exact copies of the online network and updated per-step using exponential moving average momentum $\beta_t$, scheduled by
a linear ramp from 0.985 to 0.996 during the first five epochs.  
All learnable tokens and positional embeddings are initialized with truncated normal noise ($\sigma{=}0.02$), and linear projection layers use Xavier uniform initialization.  
Training employs automatic mixed precision (\texttt{torch.amp.autocast}) with dynamic gradient scaling and explicit gradient clipping ($\|g\|_2 \!\le\! 1.0$) for stability.  
A one-epoch linear warm-up scheduler precedes cosine annealing down to $\eta_{\min}{=}10^{-5}$, implemented via a sequential scheduler stack.  
Per-batch augmentations are generated on-the-fly and differ across the two BYOL views, ensuring stochastic decorrelation even within the same mini-batch.  
Early stopping monitors validation loss improvement over five epochs, with both the best and final model checkpoints preserved for reproducibility.  
After convergence, CLS embeddings and reconstructed radar amplitudes are exported in NumPy format, followed by t-SNE and K-Means analyses automatically compiled into a PDF report containing loss curves, cluster distributions, and reconstruction plots.  

\subsubsection{Benchmark Models}
\label{subsubsec:benchmark_models}
Below, we briefly describe the benchmark models used for comparison:

\begin{itemize}
    \item \textbf{CNN baseline}~\cite{o2015introduction}: A compact convolutional encoder trained on radar–optical–climate combinations to establish a non-transformer benchmark.
    \item \textbf{SatMAE++}~\cite{cong2022satmae}: A self-supervised Transformer pretrained on Sentinel-2 data using masked autoencoding for spectral reconstruction.
    \item \textbf{ViT-S/16 (SSL4EO)}~\cite{10261879}: A Vision Transformer pretrained via contrastive learning on global Sentinel-1/2 patches, serving as a strong S1–S2 baseline.
    \item \textbf{Presto}~\cite{tseng2023lightweight}: A lightweight multimodal Transformer leveraging S1, S2, and ERA5 inputs with coordinate embeddings for geospatial alignment.
\end{itemize}


\subsubsection{Downstream Dataset}
\label{subsubsec:downstream_dataset}
We evaluate the pretrained representations on four diverse downstream datasets that differ in spatial coverage, sensing modality, and semantic complexity:
\begin{itemize}
    \item \textbf{Sickle\cite{sani2024sickle}:} Agricultural crop classification from Sentinel-2 imagery (200 samples, 5 classes).
    \item \textbf{Sen1Floods11 \cite{bonafilia2020sen1floods11}:} Flood extent detection using Sentinel-2 imagery (1,000 samples, 2 classes).
    \item \textbf{VIIRS Active Fire \cite{schroeder2014new}:} Burned area mapping combining Sentinel-1 radar and Sentinel-2 optical data (5,100 samples, 2 classes).
    % \item \textbf{GFSAD1000} \cite{Thenkabail_GFSAD_2016}: Cropland extent mapping at 1 km resolution using Sentinel-1 and Sentinel-2 data (8,000 samples, 2 classes).
    \item \textbf{Dynamic World Landcover} \cite{brown2022dynamic}: Global land cover classification from Sentinel-2, sentinel-1, climate and PlanetScope imagery with near real-time updates across 9 semantic classes (5000 samples, 9 classes).
\end{itemize}

\subsection{Results}

In this section, we present detailed quantitative results for each modality, complementing the global reconstruction metrics reported in Table~\ref{tab:pretrain_metrics_combined} of the main paper.  
All evaluations follow the same inference and normalization protocols as during pretraining.  
Metrics are reported on held-out validation data using normalized scales: optical (S2, Planet) and climate inputs were normalized to $[0,1]$, while radar (S1) values were computed on raw VV/VH amplitudes.  
Each table summarizes the coefficient of determination (R$^2$), mean squared error (MSE), mean absolute error (MAE), and F1 score for each class or feature.  
Higher R$^2$ and F1 values and lower MAE/MSE indicate improved reconstruction fidelity.


% -------------------- Sentinel-2 --------------------
\subsubsection{Sentinel-2 SpectralMAE}
Table~\ref{tab:s2_classwise} reports per-class reconstruction metrics for the Sentinel-2 SpectralMAE, illustrating consistent performance across diverse land-cover types.

\vspace{4pt}
\begin{table}[h]
\centering
\scriptsize
\caption{\textbf{Per-class reconstruction metrics for Sentinel-2 SpectralMAE.} 
All values are computed in normalized reflectance units (0–1).}
\label{tab:s2_classwise}
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{lcccc}
\toprule
\textbf{Class ID} & \textbf{R$^2$}$\uparrow$ & \textbf{F1}$\uparrow$ & \textbf{MSE}$\downarrow$ & \textbf{MAE}$\downarrow$ \\
\midrule
0 & 0.9669 & 0.9590 & 0.00029 & 0.0101 \\
1 & 0.9708 & 0.9508 & 0.00064 & 0.0112 \\
2 & 0.9872 & 0.9827 & 0.00090 & 0.0156 \\
3 & 0.9661 & 0.9639 & 0.00062 & 0.0116 \\
4 & 0.9682 & 0.9719 & 0.00119 & 0.0135 \\
5 & 0.9855 & 0.9814 & 0.00059 & 0.0117 \\
6 & 0.9668 & 0.9716 & 0.00124 & 0.0138 \\
7 & 0.9807 & 0.9763 & 0.00099 & 0.0151 \\
8 & 0.9902 & 0.9926 & 0.00124 & 0.0186 \\
\bottomrule
\end{tabular}}
\end{table}

% -------------------- Sentinel-1 --------------------
\subsubsection{Sentinel-1 BYOL + Denoising SSL}
Table~\ref{tab:s1_classwise} presents per-class results for the Sentinel-1 model trained using BYOL and denoising self-supervision.  
The model achieves high R$^2$ and F1 scores, demonstrating its ability to recover structural details from noisy radar inputs.

\vspace{4pt}
\begin{table}[h]
\centering
\scriptsize
\caption{\textbf{Per-class reconstruction metrics for Sentinel-1 BYOL+DenoisingSSL.} 
Metrics computed on raw VV/VH amplitude values.}
\label{tab:s1_classwise}
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{lcccc}
\toprule
\textbf{Class ID} & \textbf{R$^2$}$\uparrow$ & \textbf{F1}$\uparrow$ & \textbf{MSE}$\downarrow$ & \textbf{MAE}$\downarrow$ \\
\midrule
0 & 0.9957 & 0.9441 & 0.3470 & 0.3424 \\
1 & 0.9910 & 0.9910 & 0.1661 & 0.2497 \\
2 & 0.9970 & 0.9761 & 0.1825 & 0.2316 \\
3 & 0.9915 & 0.9884 & 0.3127 & 0.2978 \\
4 & 0.9872 & 0.9861 & 0.2942 & 0.2846 \\
5 & 0.9895 & 0.9857 & 0.3588 & 0.3019 \\
6 & 0.9812 & 0.9817 & 0.4438 & 0.3438 \\
7 & 0.9954 & 0.9820 & 0.3034 & 0.2802 \\
8 & 0.9857 & 0.9896 & 0.6324 & 0.4060 \\
\bottomrule
\end{tabular}}
\end{table}

% -------------------- Planet --------------------
\subsubsection{Planet SpectralMAE}
Per-class results for PlanetScope imagery are shown in Table~\ref{tab:planet_classwise}.  
Despite fewer spectral channels, the model demonstrates strong generalization with R$^2>0.95$ across most classes.

\vspace{4pt}
\begin{table}[h]
\centering
\scriptsize
\caption{\textbf{Per-class reconstruction metrics for Planet SpectralMAE.} 
All metrics are computed in normalized reflectance units (0–1).}
\label{tab:planet_classwise}
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{lcccc}
\toprule
\textbf{Class ID} & \textbf{R$^2$}$\uparrow$ & \textbf{F1}$\uparrow$ & \textbf{MSE}$\downarrow$ & \textbf{MAE}$\downarrow$ \\
\midrule
0 & 0.9224 & 0.9278 & 0.00076 & 0.0116 \\
1 & 0.9617 & 0.9331 & 0.00069 & 0.0104 \\
2 & 0.9848 & 0.9699 & 0.00077 & 0.0122 \\
3 & 0.9305 & 0.9349 & 0.00065 & 0.0108 \\
4 & 0.9497 & 0.9227 & 0.00051 & 0.0096 \\
5 & 0.9727 & 0.9536 & 0.00045 & 0.0095 \\
6 & 0.9535 & 0.9372 & 0.00049 & 0.0098 \\
7 & 0.9876 & 0.9765 & 0.00048 & 0.0113 \\
8 & 0.9869 & 0.9839 & 0.00167 & 0.0224 \\
\bottomrule
\end{tabular}}
\end{table}

% -------------------- Climate --------------------
\subsubsection{Climate MAE}
Table~\ref{tab:climate_classwise} reports per-class results for the Climate MAE.  
The model attains high R$^2$ and F1 for temperature-related variables, while precipitation exhibits expected variability due to its higher spatiotemporal noise.

\vspace{4pt}
\begin{table}[h]
\centering
\scriptsize
\caption{\textbf{Per-class reconstruction metrics for Climate MAE.} 
Metrics are computed in standardized feature space.}
\label{tab:climate_classwise}
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{lcccc}
\toprule
\textbf{Class ID} & \textbf{R$^2$}$\uparrow$ & \textbf{F1}$\uparrow$ & \textbf{MSE}$\downarrow$ & \textbf{MAE}$\downarrow$ \\
\midrule
0 & 0.8565 & 0.9865 & 0.0148 & 0.0289 \\
1 & 0.7595 & 0.9529 & 0.0280 & 0.0540 \\
2 & 0.9740 & 0.9769 & 0.0095 & 0.0304 \\
3 & 0.8735 & 0.9763 & 0.0074 & 0.0186 \\
4 & 0.9703 & 0.9776 & 0.0012 & 0.0143 \\
5 & 0.9248 & 0.9811 & 0.0117 & 0.0376 \\
6 & 0.8785 & 0.9655 & 0.0089 & 0.0257 \\
7 & 0.9850 & 0.9828 & 0.0043 & 0.0300 \\
8 & 0.8712 & 0.9964 & 0.0746 & 0.1066 \\
\bottomrule
\end{tabular}}
\end{table}

\noindent These extended quantitative evaluations highlight the stability and cross-sensor robustness of the pretrained models, complementing the global metrics summarized in Table~\ref{tab:pretrain_metrics}.

\paragraph{Spectral Reconstruction Visualization.}
To qualitatively assess reconstruction fidelity, we visualize the original and reconstructed reflectance curves for representative samples from each land cover class across modalities. 
For every modality, one example per class was selected (middle index within each class), and the predicted spectral response was plotted against the true reflectance values across all input bands. 
This allows direct comparison of band-wise reconstruction accuracy and highlights how each model preserves spectral smoothness and relative band intensities.
Figure~\ref{fig:reconstruction_grid} illustrates these reconstructions for Sentinel-2, PlanetScope, and Sentinel-1, demonstrating strong correlation between original and reconstructed values across diverse surface types.

\vspace{7pt}
\begin{figure*}[htbp]
    \centering
    % --- Sentinel-2 ---
    \begin{subfigure}[b]{0.33\textwidth}
        \includegraphics[width=\linewidth]{img/S2_reconstruction_grid.png}
        \caption{Sentinel-2 Spectral-MAE}
    \end{subfigure}
    \hfill
    % --- Planet ---
    \begin{subfigure}[b]{0.33\textwidth}
        \includegraphics[width=\linewidth]{img/Planet_reconstruction_grid.png}
        \caption{Planet MAE}
    \end{subfigure}
    \hfill
    % --- Sentinel-1 ---
    \begin{subfigure}[b]{0.33\textwidth}
        \includegraphics[width=\linewidth]{img/S1_reconstruction_grid.png}
        \caption{Sentinel-1 BYOL+Denoise}
    \end{subfigure}

    \vspace{4pt} % space between images and caption
    \caption{\textbf{Original vs reconstructed reflectance (per class).} 
    Each subplot compares the true (solid) and reconstructed (dashed) reflectance curves for a representative pixel from every land cover class. 
    The close alignment indicates accurate per-band reconstruction and strong spectral consistency across modalities.}
    \label{fig:reconstruction_grid}
\end{figure*}
\vspace{6pt}

\paragraph{t-SNE Visualization of Latent Representations.}
To better understand the structure of the learned embeddings, we visualize the 2D t-SNE projections of the latent representations obtained from each pretrained encoder. 
A subset of 10,000 embeddings per modality was sampled and clustered using \textit{k}-means with $K=9$ to reveal semantic separability among different land cover types. 
The resulting t-SNE scatter plots show how the models organize samples with similar physical or spectral characteristics into coherent clusters, highlighting the discriminative power of the self-supervised embeddings.
Figure~\ref{fig:tsne_grid} compares the t-SNE projections across Sentinel-2, PlanetScope, and Sentinel-1 modalities.

\vspace{7pt}
\begin{figure*}[htbp]
    \centering
    % --- Sentinel-2 ---
    \begin{subfigure}[b]{0.33\textwidth}
        \includegraphics[width=\linewidth]{img/S2_tsne.png}
        \caption{Sentinel-2 Spectral-MAE}
    \end{subfigure}
    \hfill
    % --- Planet ---
    \begin{subfigure}[b]{0.33\textwidth}
        \includegraphics[width=\linewidth]{img/Planet_tsne.png}
        \caption{Planet MAE}
    \end{subfigure}
    \hfill
    % --- Sentinel-1 ---
    \begin{subfigure}[b]{0.33\textwidth}
        \includegraphics[width=\linewidth]{img/S1_tsne.png}
        \caption{Sentinel-1 BYOL+Denoise}
    \end{subfigure}

    \vspace{4pt} % space between images and caption
    \caption{\textbf{t-SNE visualization of modality-specific embeddings.} 
    Each plot shows the two-dimensional t-SNE projection of 10,000 embeddings, color-coded by $K$-means cluster ID ($K=9$). 
    The clear clustering patterns indicate that the learned latent spaces effectively capture modality-relevant semantic structure and inter-class relationships.}
    \label{fig:tsne_grid}
\end{figure*}
\vspace{6pt}


% --------------------------------anjali---------------------------
% ========================
% 7.1 Embedding Reconstruction
% ========================
\subsection{Embedding Reconstruction on Different Land Cover Types}

To assess the representational fidelity of the learned embeddings, we reconstruct imagery from embeddings for different land cover types-\textit{urban}, \textit{vegetation}, \textit{water}, and \textit{snow} regions-using sentinel-2, sentinel-1 and Planet specific model.

\paragraph{PCA-based Reconstruction Approach.}
Following the approach proposed in the \textit{TESSERA} paper, we perform a dimensionality reduction of the learned embedding vectors via Principal Component Analysis (PCA).  
The first three principal components (PC1, PC2, and PC3) are mapped to the RGB color channels to form a pseudo-RGB visualization.  
This enables interpretable spatial visualization of the embedding structure without retraining the decoder.  
Comparing these reconstructions with ground truth inputs provides insight into the degree to which spectral-spatial information is preserved in the latent space.



% -----------------------------------------------------
\subsubsection{Sentinel-2 (S2) Embedding Reconstruction}
A qualitative overview of the reconstructed Sentinel-2 (S2) embeddings can be observed in Figure~\ref{fig:s2_reconstruction}

\begin{figure*}[t!]
    \centering
    % --- Water ---
    \begin{subfigure}[b]{0.252\textwidth}
        \includegraphics[width=\linewidth]{img/s2_water_bodies.png}
        \caption{Coastal Water Bodies - GT}
    \end{subfigure}
    \begin{subfigure}[b]{0.252\textwidth}
        \includegraphics[width=\linewidth]{img/s2_water_bodies_recon.png}
        \caption{Reconstructed (S2)}
    \end{subfigure}
    % --- Snow ---
    \begin{subfigure}[b]{0.24\textwidth}
        \includegraphics[width=\linewidth]{img/s2_snow.png}
        \caption{Snow - GT}
    \end{subfigure}
    \begin{subfigure}[b]{0.24\textwidth}
        \includegraphics[width=\linewidth]{img/s2_snow_recon.png}
        \caption{Reconstructed (S2)}
    \end{subfigure}

    \vspace{0.8em}

    % --- Urban ---
    \begin{subfigure}[b]{0.252\textwidth}
        \includegraphics[width=\linewidth]{img/s2_urban.png}
        \caption{Urban - GT}
    \end{subfigure}
    \begin{subfigure}[b]{0.252\textwidth}
        \includegraphics[width=\linewidth]{img/s2_urban_recon.png}
        \caption{Reconstructed (S2)}
    \end{subfigure}

    \caption{Sentinel-2 embedding reconstructions for different land cover types using PCA-mapped embeddings. Ground truth (GT) vs reconstructed representations are shown side-by-side.}
    \label{fig:s2_reconstruction}
\end{figure*}



% -----------------------------------------------------

\begin{figure*}[t!]
    \centering
    \begin{subfigure}[b]{0.252\textwidth}
        \includegraphics[width=\linewidth]{img/s1_water_bodies.png}
        \caption{Coastal Water Bodies - GT}
    \end{subfigure}
    \begin{subfigure}[b]{0.252\textwidth}
        \includegraphics[width=\linewidth]{img/s1_water_bodies_recon.png}
        \caption{Reconstructed (S1)}
    \end{subfigure}
    \begin{subfigure}[b]{0.223\textwidth}
        \includegraphics[width=\linewidth]{img/s1_snow.png}
        \caption{Snow - GT}
    \end{subfigure}
    \begin{subfigure}[b]{0.223\textwidth}
        \includegraphics[width=\linewidth]{img/s1_snow_recon.png}
        \caption{Reconstructed (S1)}
    \end{subfigure}

    \vspace{0.8em}

    \begin{subfigure}[b]{0.24\textwidth}
    \includegraphics[width=\linewidth, height=0.45\textwidth]{img/s1_grassland.png}
    \caption{Urban - GT}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
    \includegraphics[width=\linewidth, height=0.45\textwidth]{img/s1_grassland_recon.png}
    \caption{Reconstructed (S1)}
\end{subfigure}


    \caption{Sentinel-1 embedding reconstructions for different land cover types using PCA-mapped embeddings. Ground truth (GT) vs reconstructed representations are shown side-by-side.}
    \label{fig:s1_reconstruction}
\end{figure*}


% -----------------------------------------------------


\begin{figure*}[t!]
    \centering
    \begin{subfigure}[b]{0.25\textwidth}
        \includegraphics[width=\linewidth]{img/planet_water_bodies.png}
        \caption{Coastal Water Bodies - GT}
    \end{subfigure}
    \begin{subfigure}[b]{0.25\textwidth}
        \includegraphics[width=\linewidth]{img/planet_water_bodies_recon.png}
        \caption{Reconstructed (Planet)}
    \end{subfigure}
    \begin{subfigure}[b]{0.212\textwidth}
        \includegraphics[width=\linewidth]{img/planet_snow.png}
        \caption{Snow - GT}
    \end{subfigure}
    \begin{subfigure}[b]{0.212\textwidth}
        \includegraphics[width=\linewidth]{img/planet_snow_recon.png}
        \caption{Reconstructed (Planet)}
    \end{subfigure}

    \vspace{0.8em}

    \begin{subfigure}[b]{0.20\textwidth}
        \includegraphics[width=\linewidth]{img/planet_urban.png}
        \caption{Urban - GT}
    \end{subfigure}
    \begin{subfigure}[b]{0.20\textwidth}
        \includegraphics[width=\linewidth]{img/planet_urban_recon.png}
        \caption{Reconstructed (Planet)}
    \end{subfigure}

    \caption{Planet embedding reconstructions for different land cover types using PCA-mapped embeddings. Ground truth (GT) vs reconstructed representations are shown side-by-side.}
    \label{fig:planet_reconstruction}
\end{figure*}

\subsubsection{Sentinel-1 (S1) Embedding Reconstruction}
The results of the Sentinel-1 (S1) embedding reconstruction are illustrated in Figure~\ref{fig:s1_reconstruction}.


\subsubsection{PlanetScope Embedding Reconstruction}
The results of the PlanetScope embedding reconstruction are illustrated in Figure~\ref{fig:planet_reconstruction}.



% ---------------------------------------------sir
\subsubsection{Summary of Key Innovations}

\begin{table*}[t]
\centering
\begin{threeparttable}
\caption{\textbf{Summary of Key Innovations.}
SPEAR introduces architectural innovations for pixel-level multi-modal spectral fusion in self-supervised Earth observation.}
\label{tab:innovations}

\small
\renewcommand{\arraystretch}{1.2}
\setlength{\tabcolsep}{4pt}

\begin{tabular}{c
                p{0.22\textwidth}
                p{0.33\textwidth}
                p{0.23\textwidth}}
\toprule
\# & \textbf{Novel Component} & \textbf{Description} & \textbf{Impact} \\
\midrule

1 & Pixel-wise SSL &
Learns per-pixel latent spectra &
Fine-grained precision \\

2 & $\lambda$--$\Delta\lambda$ Meta-Embedding &
Wavelength-aware spectral tokens &
Sensor-agnostic reasoning \\

3 & Dual SSL (MAE + BYOL) &
Joint reconstruction + denoising &
Noise robustness \\

4 & Sensor \& GSD Encoding &
Sensor identity + ground sampling encoding &
Cross-domain fusion \\

5 & Pixel-level Fusion Transformer &
Aligns optical, radar, and climate embeddings &
Robust to missing sensors \\

6 & Unified SSL → Downstream &
Single encoder reused for all stages &
Efficient training + inference \\

7 & Modality Dropout Robustness &
Invariance to missing or corrupted inputs &
Stable multi-sensor deployment \\

8 & EO-wide Pretraining (S1+S2+Planet+ERA5) &
Multi-physics, multi-modal representation learning &
High transferability \\

\bottomrule
\end{tabular}

\end{threeparttable}
\end{table*}



%-------------------------------------------
