🔬 physics

Multi-Fidelity Emulation of Atmospheric Correction Coefficients with Physics-Guided Kolmogorov-Arnold Networks

This paper introduces pKANrtm, a physics-guided Kolmogorov-Arnold Network that leverages multi-fidelity emulation to accurately and efficiently predict high-fidelity atmospheric correction coefficients by correcting low-fidelity 6S simulations with libRadtran residuals, achieving significant computational speedups over traditional radiative transfer models.

Original authors: Md Abdullah Al Mazid, Naphtali Rishe

Published 2026-05-13

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Md Abdullah Al Mazid, Naphtali Rishe

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to bake the perfect cake. To do this, you need to know exactly how the oven, the ingredients, and the humidity in the kitchen will affect the final result. In the world of satellite imaging, "baking the cake" is called atmospheric correction. Satellites look down at Earth, but the air between the satellite and the ground (the atmosphere) distorts the picture. It adds haze, absorbs colors, and scatters light. To get a clear, true picture of the ground, scientists have to mathematically "remove" the atmosphere's effect.

The problem is that calculating exactly how the atmosphere distorts light is like trying to solve a massive, complex physics puzzle for every single pixel in an image. Doing this with the most accurate tools is incredibly slow and expensive for computers, like trying to bake a cake by hand-cranking a mixer for every single batch.

Here is how this paper solves that problem, using simple analogies:

1. The "Fast Approximation" vs. The "Slow Masterpiece"

Scientists usually use two types of tools to solve this puzzle:

The "Fast Approximation" (6S): Think of this as a quick, experienced baker who can guess the cake's outcome in a split second. It's fast, but it's not 100% perfect.
The "Slow Masterpiece" (libRadtran): This is the master chef who measures every gram of flour and every degree of heat. The result is perfect, but it takes hours to bake just one cake.

The paper's goal was to figure out how to get the Master Chef's perfect results without waiting hours for them.

2. The "Smart Assistant" (The AI Model)

The researchers built a new AI assistant called pKANrtm. Instead of trying to learn the whole recipe from scratch, this assistant uses a clever trick called "Multi-Fidelity Emulation."

Here is how it works:

The Setup: The assistant watches the "Fast Approximation" (6S) bake the cake first.
The Prediction: The assistant doesn't try to predict the final cake directly. Instead, it predicts the difference (the "residual") between what the Fast Approximation guessed and what the Master Chef would have actually made.
The Physics Check: The assistant is also given a rulebook (physics-guided). It's told, "You can make a guess, but it must follow the laws of physics." If the assistant predicts something that breaks the laws of nature (like a cake that floats), it gets penalized. This ensures the AI stays grounded in reality.

3. The "Spectral Response" (The Colored Glasses)

Satellites don't see all colors equally; they see specific "bands" of light through special filters (like wearing different colored glasses). The paper made sure their AI learned to look through these specific "glasses" (Sentinel-2 bands) so the results would match exactly what the satellite sees, rather than just a generic guess.

4. The Results: Speed and Accuracy

The researchers tested this new AI assistant against other methods and found:

Accuracy: The assistant was incredibly accurate. It could predict the "Master Chef's" perfect results almost as well as the Master Chef itself, even when the weather conditions were weird or different from what it had seen before (like testing it on a rainy day when it mostly trained on sunny days).
Speed: This is the big win.
- The "Master Chef" (libRadtran) took about 37 seconds to calculate one sample.
- The AI assistant on a standard computer took about 0.1 seconds.
- The AI assistant on a powerful graphics card (GPU) took about 0.003 seconds.
- The Analogy: If the Master Chef takes 37 seconds to bake one cake, the AI assistant can bake 48,000 cakes in that same second when working in a batch. It's a speedup of roughly 11,000 times.

5. Where It Struggles

The paper notes that the AI is great at most things, but it still finds it a little hard to handle "absorption-sensitive" bands. Imagine trying to see through a thick fog or a very dark stain; the AI sometimes struggles to predict exactly how much light is lost in those specific, tricky areas. However, for the vast majority of the image, it works perfectly.

Summary

In short, this paper presents a new, super-fast AI assistant that learns to fix satellite images. It doesn't replace the slow, perfect physics tools; instead, it learns to predict the difference between a fast, rough guess and the perfect answer. By doing this, it allows scientists to process satellite data thousands of times faster while keeping the results physically accurate and reliable.

Technical Summary: Multi-Fidelity Emulation of Atmospheric Correction Coefficients with Physics-Guided Kolmogorov-Arnold Networks

Problem Statement
Atmospheric correction is a critical preprocessing step in optical remote sensing, requiring the removal of atmospheric effects (scattering, absorption) from Top-of-Atmosphere (TOA) radiance to retrieve surface reflectance. Physics-based Radiative Transfer Models (RTMs) like 6S, MODTRAN, and libRadtran are the standard for this task. However, repeated execution of high-fidelity RTMs is computationally expensive, creating bottlenecks for dense Look-Up Table (LUT) generation, sensitivity analysis, and operational processing. While surrogate modeling (emulators) has been proposed to accelerate these workflows, challenges remain regarding accuracy, extrapolation to out-of-distribution (OOD) states, and maintaining physical consistency. Furthermore, many existing emulators predict surface reflectance directly, bypassing the intermediate physical coefficients used in standard correction equations.

Methodology
The authors propose a physics-aware, multi-fidelity surrogate framework to emulate atmospheric correction coefficients rather than raw reflectance. The methodology consists of the following components:

Multi-Fidelity Data Generation:
- Low-Fidelity (LF) Model: 6S, used for its computational efficiency.
- High-Fidelity (HF) Model: libRadtran (using the DISORT solver), serving as the accurate target.
- Sampling: Latin Hypercube Sampling (LHS) is used to generate 50,000 unique atmospheric-geometric states (covering Aerosol Optical Depth, Column Water Vapour, ozone, solar/view geometry, and elevation).
- Spectral Integration: Simulations are performed for 13 Sentinel-2 bands using Spectral Response Function (SRF) awareness to ensure coefficients match sensor-level observations.
- Target Variables: The study focuses on three atmospheric correction coefficients: path reflectance ( $\rho_{path}$ ), total transmittance ( $T_{total}$ ), and spherical albedo ( $s$ ).
- Quality Control: A rigorous filtering process removes non-finite values and physically implausible coefficients, resulting in a final dataset of ~610,000 paired samples.
Model Architecture (pKANrtm):
- The core model is a Physics-guided Kolmogorov-Arnold Network (pKANrtm).
- Formulation: The problem is framed as residual multi-fidelity learning. The network takes the atmospheric state vector ( $x$ ) and the low-fidelity 6S coefficients ( $y_{LF}$ ) as input. It predicts the residual ( $r = y_{HF} - y_{LF}$ ) required to correct the 6S output to match libRadtran.
- Architecture: An Efficient-KAN implementation with two hidden layers (widths 256 and 128). Unlike standard MLPs, KANs use learnable univariate functions on edges.
- Physics-Guided Training: The loss function combines standard Mean Squared Error (MSE) on standardized residuals with a physics-consistency penalty ( $\lambda_{phys} P_{phys}$ ). This penalty is applied in the original, unscaled coefficient space to discourage predictions that violate physical constraints, even if they appear small in standardized error space.
Evaluation Strategy:
- Splits: Data is split at the state level (not row level) to prevent leakage. Two regimes are tested: a standard split (interpolation) and an OOD split (focusing on high AOD and high CWV states).
- Comparators: The model is benchmarked against sRTMNet (hybrid RTM-neural), Random Forests, Fully Connected networks, Gaussian Processes, and Denoising Autoencoders.
- Metrics: RMSE, MAE, $R^2$ , and SMAPE are computed on the reconstructed high-fidelity coefficients.

Key Results

Predictive Performance: pKANrtm achieved the best overall performance among all compared models.
- Standard Split: RMSE = 0.00619, $R^2$ = 0.99472. It outperformed the sRTMNet baseline by reducing RMSE by ~10.5% and SMAPE by ~8.6%.
- OOD Split: RMSE = 0.01022, $R^2$ = 0.99418. Despite the distribution shift, the model maintained high variance capture, outperforming the sRTMNet baseline by ~12.2% in RMSE.
Band-wise Analysis: Most Sentinel-2 bands (visible, red-edge, NIR) were accurately emulated. However, absorption-sensitive bands (particularly B9 and B10) remained challenging, with B10 showing the highest error due to strong atmospheric absorption and fewer valid samples after quality control.
Computational Efficiency:
- libRadtran CPU execution: ~37.33 seconds per sample.
- pKANrtm CPU inference: ~~95.7 ms per sample (~~390x speedup).
- pKANrtm GPU inference (single sample): ~~3.4 ms (~~11,000x speedup).
- Batched GPU Inference: Achieved a throughput of ~48,400 samples per second.

Significance and Claims
The paper claims that the pKANrtm framework offers a practical pathway for fast, accurate, and physically structured atmospheric correction coefficient generation. Its significance lies in three main areas:

Coefficient-Level Emulation: By emulating the intermediate coefficients ( $\rho_{path}$ , $T_{total}$ , $s$ ) rather than direct reflectance, the approach preserves the interpretability and compatibility of standard Lambertian atmospheric correction equations.
Physics-Guided Residual Learning: The combination of multi-fidelity learning (using 6S as a guide) and physics-consistency penalties allows the model to learn complex, state-dependent corrections while adhering to physical constraints, resulting in superior performance over purely data-driven baselines.
Operational Scalability: The massive acceleration provided by GPU inference makes the generation of dense LUTs and large-scale sensitivity analyses computationally feasible, addressing a primary bottleneck in operational remote sensing workflows.

The authors acknowledge limitations, noting that the evaluation relies on synthetic RTM simulations rather than field-measured ground truth, and that absorption-sensitive bands (like B10) require further specialized handling. The dataset and code are publicly released to support reproducibility and future benchmarking.