Manuscript, code and run records for a comparison of neural networks, exact Matérn kernel regression and their combinations as emulators of a six-dimensional radiative-transfer table on the 285-band EMIT wavelength grid, and for an analysis of how their forward accuracy carries over to reflectance retrieval.
Paper (PDF) · Supplement (PDF) · LaTeX source · How each table is produced
On ten partitions of 23,313 tabulated states, each with 18,884 training rows and validation-based selection, a Matérn kernel with one length scale per input reaches 0.095% mean relative radiance error, against 0.760% for cubic regression and 0.387% for a 3 × 512 network. A residual kernel brings the network to 0.141%. An exact kernel on the hidden features of a width-2,000 network reaches 0.079%, and one kernel on the concatenated features of five width-512 networks reaches 0.076%.
Forward and inverse rankings disagree. The convex stack built on the width-2,000 network improves on its own feature kernel by 0.0006 radiance percentage points, while its mean within-split 95th-percentile reflectance error is 22.71 points against 7.57, larger at every partition. On the physical domain of the retrieval, scored with its coverage and failed inversions, the order persists at a smaller ratio: at the transmission floor 1e-12 the stack's 95th percentile is 1.6 to 1.9 times the feature kernel's at width 512, and the ratio is above one at every floor, on every split. On a fresh partition evaluated once under a hypothesis fixed in advance, the ratio is 6.60 over all bands and 5.54 on the screened entries.
The analysis proves that on the physical domain the constrained retrieval error is at most the retrieval-weighted component error divided by the transmission, with constant one; that the rate this implies is sharp; and that component errors proportional to the transmission lose no rate. Applied band by band, the bound marks in advance the bands where the retrieval error is large.
Retraining every family only on physically admissible targets lowers the albedo error of every family and, at both widths, the radiance error of the feature kernel and the stack (the wide stack to 0.061%); the network itself gets worse at width 512 and is mixed at width 2000. At width 512 it also raises the 95th-percentile retrieval error on the physical domain for every family on average, and for each family on at least 8 of 10 splits: the filter moves component error toward the smallest transmissions, where the retrieval divides by them.
On a second radiative-transfer code, libRadtran over the thirteen Sentinel-2 bands (9,722 states, ten splits), the reversal is much weaker: the hypothesis that the stack is no worse on radiance and worse in the tail holds on 5 of 10 splits, and the stack's all-band tail is 0.92 to 1.13 times the feature kernel's, against 2.6 to 4.9 on the EMIT table. Away from band B10, which is nearly opaque and where the remaining difference lies, every transmission exceeds 0.14, so the division by the transmission amplifies component error by at most about seven times.
paper/: the article (emit_kernel_dnn.tex), its supplement (supplement.tex: the proofs, the further experiments and the extended tables), the tables of both (generated from the records) and the two compiled PDFs.code/: drivers, scoring, table and figure scripts, and tests.code/lanes/holds the scripts that ran each fit of the training-target experiment, the libRadtran comparison and the correction-coefficient benchmark.results/: one JSON record per run, with the digests of its data and split indices.results/target_quality/andresults/libradtran/also hold the protocols of those two experiments with the SHA-256 of each, recorded before the first fit;results/confirmation/holds the protocol, the frozen hyperparameters with their SHA-256, the scripts and the report of the fresh-partition evaluation.figures/: the figures of the paper.
With Python 3.11 or later and a TeX installation:
python -m pip install -r requirements.txt
python -m unittest discover -s code -p 'test_*.py' -v
python code/make_width_numbers.py
python code/make_scaling_tables.py
python code/make_retune_table.py
python code/make_emit_tables.py
python code/make_tables.py
python code/make_v2_tables.py
python code/format_publication_tables.py
python code/make_domain_table.py
python code/make_tq_tables.py
python code/make_lrt_tables.py
python code/make_lrt_tables.py lrt
python code/make_confirmation_table.py
(cd paper && pdflatex -interaction=nonstopmode supplement.tex)
latexmk -pdf -interaction=nonstopmode -halt-on-error -cd paper/emit_kernel_dnn.tex
latexmk -g -pdf -interaction=nonstopmode -halt-on-error -cd paper/supplement.texThese commands run the tests and regenerate every table of the paper from the records in results/, byte for byte;
they do not retrain any model. docs/REPRODUCE.md maps every table and figure to its records
and scripts and gives the commands that redraw the figures and rerun the training.
The EMIT table was provided by the Jet Propulsion Laboratory for EMIT atmospheric correction and is identified in the
paper by size and SHA-256; the arrays are available from the author on request, subject to JPL's permission. The
libRadtran arrays are built from the public release of the paired 6S/libRadtran Sentinel-2 corpus with
code/make_libradtran_arrays.py. The OCO-2 arrays and reference predictions come from the release accompanying
Lamminpää et al. (2025); the other corpora are the cited public releases.
Claude Code (Anthropic) and Codex (OpenAI) implemented and ran the experiments and drafted the manuscript; ChatGPT (OpenAI) developed parts of the analysis. Funding and competing interests are stated in the paper. The companion method and experiments are in neural-means-kernel-corrections. The MIT license covers the code in this repository, not rights to data that are not distributed here.