CREST-Global · Stage-1 Snow-17

Headline

One differentiable model, one 15-year record, one continental parameter field — with an honest read on where it works and why.

What it is

A per-cell Snow-17 parameter set (9 params) that reproduces SNODAS SWE, learned by gradient descent rather than SCE-UA/DREAM. Grid-independent, so it scales toward a global product.

How good

Timing is captured everywhere (r 0.71–0.92). Continental regions reach business-grade skill (KGE 0.60–0.73). The residual is bias, concentrated in maritime deep snow.

Key idea

SWE-only data leaves the melt/retention parameters weakly identified. A physical prior regularizes them to literature values — reasonable and accurate.

Accuracy by region

Evaluated on a held-out temporal test period (last third of the record), over seasonal-snowpack cells (peak SWE ≥ 100 mm). KGE is decomposed into r (timing), α (variability ratio), β (mean/bias ratio).

Region	KGE	r (r²)	α	β (bias)	read
Great Lakes / Upper Midwest	0.69	0.91 (0.82)	0.83	0.85	strongest — cold, lake-effect
Northeast / Appalachia	0.68	0.86 (0.73)	0.98	0.91	near-unbiased
Mid / South Rockies	0.60	0.83 (0.69)	0.99	1.04	best mountains (β≈1)
North Rockies	0.60	0.82 (0.68)	0.97	0.86	good
Southwest high	0.47	0.77 (0.60)	0.92	0.91	small sample
Wasatch / Great Basin	0.41	0.77 (0.59)	1.04	1.26	over-sim, desert margin
Sierra Nevada	0.43	0.76 (0.58)	0.81	0.84	hard — maritime deep snow
Cascades	0.28	0.68 (0.47)	0.90	0.73	hardest — undercatch

Read: the model gets the accumulation–melt timing right almost everywhere (high r). What varies is bias (β): continental basins sit at β≈0.85–1.05; maritime deep-snow basins under-simulate (β 0.73–0.84). The ceiling is not the calibration — it is forcing/truth representativeness in maritime terrain.

Per-region accuracy: KGE bar chart, sim-vs-obs scatter, region-mean SWE time series — (A) median test-KGE by region; (B/C) sim-vs-SNODAS SWE density scatter; (D) region-mean SWE, observed vs simulated (grey band = temporal test period).

Final CONUS maps of test-KGE and key Snow-17 parameters — Final CONUS product: test-KGE over the seasonal snowpack, plus the calibrated SCF, PXTEMP, MFMAX, MFMIN fields and the applied temperature correction.

Why calibrate at all — default vs learned parameters

Snow-17 ships with Anderson's classic default parameters (MFMAX 1.4, MFMIN 0.3, SCF 1.1, PXTEMP 1.0, …). A fair head-to-head: identical forcing (including the temperature correction), identical model — only the parameters differ.

Region	per-cell median KGE — default	per-cell median KGE — calibrated	Δ
Northeast / Appalachia	−0.12	+0.83	+0.95
Great Lakes / Upper Midwest	+0.15	+0.79	+0.64
Mid / South Rockies	+0.10	+0.63	+0.53
North Rockies	+0.14	+0.62	+0.48
Southwest high	+0.19	+0.58	+0.39
Cascades	−0.12	+0.49	+0.61
Wasatch / Great Basin	−0.74	+0.48	+1.22
Sierra Nevada	−0.13	+0.45	+0.58

Read: with one nationwide default parameter set, half the regions score negative per-cell KGE — worse than predicting the mean seasonal cycle. Backprop calibration lifts every region to +0.45…+0.83 (average gain ≈ +0.6 KGE). Region-mean SWE is more forgiving of defaults (continental dry-snow regions average out to KGE 0.77–0.89 even uncalibrated), but grid-cell-level skill — what a distributed model is for — requires calibration.

Default vs calibrated Snow-17: time series and per-cell KGE bars — Top: region-mean SWE for four regions — SNODAS observations (blue fill), Snow-17 with default parameters (grey dashed; peaks too high, melt-out too late), and backprop-calibrated (red). Bottom: per-cell median KGE by region, default vs calibrated.

Where the regions are, and how each tracks the truth

First: the eight assessment regions on the map. Then, for each region, the region-mean SWE — observed (SNODAS) vs simulated (Snow-17) — split into the training years and the held-out test years, side by side.

Map of the eight assessment regions over CONUS — The eight regions, drawn over SNODAS peak-SWE shading (darker = deeper snow). Boxes are the exact bounding boxes used for every number on this page.

Two different KGE numbers — both honest. The region table above reports the per-cell median KGE (how a typical single cell does — harder, includes per-cell noise). The time-series below report the region-mean KGE (how the aggregated regional snowpack does — noise averages out, so these are higher: 0.54–0.95). Aggregate skill is what matters for basin water supply; per-cell skill is what matters for spatial detail.

Per-region time series — train vs test (split at 2019-12-29)

Northeast SWE train vs test — Northeast / Appalachia — region-mean KGE train 0.90 · test 0.94.

Great Lakes SWE train vs test — Great Lakes / Upper Midwest — KGE train 0.87 · test 0.83.

Mid/S Rockies SWE train vs test — Mid / South Rockies — KGE train 0.80 · test 0.95.

Wasatch SWE train vs test — Wasatch / Great Basin — KGE train 0.86 · test 0.92 (note the mild over-simulation of peaks).

N Rockies SWE train vs test — North Rockies — KGE train 0.71 · test 0.80.

Southwest high SWE train vs test — Southwest high — KGE train 0.76 · test 0.73.

Sierra SWE train vs test — Sierra Nevada — KGE train 0.60 · test 0.73; peaks slightly under-simulated (maritime deep snow).

Cascades SWE train vs test — Cascades — KGE train 0.60 · test 0.54; the hardest region (under-catch + sub-grid orography).

How to read these numbers

Is r² = 0.9 good? For SWE, not by itself. Here is why we lead with KGE.

r² / r — necessary, not sufficient

SWE has a huge, smooth seasonal cycle (0 → peak → 0 each year). Any model that tracks that cycle scores r ≈ 0.8–0.9 automatically. So high r² only confirms timing; it says nothing about whether peak magnitude or total volume is right, and it does not penalize bias.

KGE — the honest metric

KGE = 1 − √[ (r−1)² + (α−1)² + (β−1)² ]

Requires timing (r), the right amplitude (α = σ_sim/σ_obs), and the right mean (β = μ_sim/μ_obs). Cascades has r = 0.68 but KGE only 0.28 — because β = 0.73 (27% under-bias). r² would hide that; KGE surfaces it.

Benchmarks

KGE	meaning
1.00	perfect
≥ 0.75	good
0.50 – 0.75	acceptable / behavioural
> −0.41	beats the mean-of-observations benchmark (the true "zero line", not 0)

For a globally-transferable 0.1° product with no per-station tuning, driven by global reanalysis and validated out-of-time, continental KGE 0.6–0.73 with r 0.8–0.92 is a solid, competitive result.

Methods — accurate and reasonable parameters

The order of operations: verify the forcing, diagnose why parameters rail at their bounds, then regularize — not swap inputs blindly.

1 · The forcing is already good

Temperature is unbiased. ERA5-Land minus SNOTEL melt-season air temperature (lapse-corrected to grid elevation) = −0.04 °C median across 825 stations. No temperature swap needed.
Precipitation is adequate at grid scale. Peak-SWE / cold-season-snowfall ratio ≈ 0.36; only ~5% of cells could not build the observed pack from ERA5. SNODAS / SNOTEL peak = 0.81 (grid-mean vs point — expected, not broken).

2 · Why parameters railed — a weak-identifiability test

In the baseline fit, MFMAX/MFMIN pinned at their upper bounds and PLWHC at its lower bound. We widened the bounds drastically (MFMAX→5, MFMIN→1.5, SCF→0.4). Result: test-KGE moved only +0.05–0.10 while the parameters ran to the new extremes. Conclusion: the melt/retention parameters are weakly identified under an SWE-only loss — the likelihood surface is flat, so they drift to whatever bound they are given. The bounds were never the limit.

3 · The fix — a physical prior (Tikhonov / MAP)

Where the data cannot constrain a parameter, inject domain knowledge. We add an L2 penalty pulling the weakly-identified parameters toward literature-central values, in normalized units:

L = L_data(SWE) + 2·L_spatial(smoothness) + λ · Σ_k mean[ ((θ_k − target_k)/(hi_k−lo_k))² ]

λ is the regularization weight (a Gaussian prior with λ ∝ 1/σ²_prior). We use λ = 200.
Applied only to weakly-identified params: MFMAX, MFMIN, NMF, PLWHC, TIPM, UADJ.
SCF and PXTEMP are deliberately not regularized — they are identifiable, so the data speaks.
Cost: continental median test-KGE 0.51 → 0.47 (−0.036). Benefit: bound-railing MFMAX 54%→0.8%, MFMIN 76%→3%, NMF 59%→2%. Spatial signal is preserved (MFMAX retains real structure; SCF keeps corr 0.64 with peak SWE).

Honesty note: the prior target is the empirical operational value from decades of Snow-17 studies, not a physical law and not the midpoint of the range. It is justified because (a) the parameters are provably weakly identified and (b) the forcing is bias-free, so little compensation is being erased. Under biased forcing this prior would not be appropriate.

Baseline vs physical-prior parameter maps — Baseline (left) vs physical-prior λ=200 (right). MFMAX/MFMIN move from saturated bounds to physical values while keeping spatial structure; TIPM tile-seam artifacts vanish; SCF (data-driven) is unchanged.

4 · Precipitation swap — tested, low yield here

Would a "better" precip product fix the maritime bias? Evidence says no: Sierra ERA5 precip is already ~right (0.96× SNOTEL) yet SWE is under-biased with SCF < 1 (the model is shedding precip, not starved of it). SCF is itself a free per-cell precip scaler, so uniform amount errors are already absorbed — the Rockies undercatch 23% yet sit at β≈1. The residual maritime bias looks like sub-grid orographic representativeness and rain/snow partition, which a precip swap cannot resolve. (IMERG was unusable — the CONUS cube was 98.6% NaN; MSWEP needs manual download.)

Parameters, ranges & priors

The nine Snow-17 parameters, their search ranges, and the prior target (literature-central value) where one is applied.

Param	Meaning	Range	Prior target	Regularized?	Identifiable?
scf	precip / snow correction (undercatch)	0.7–1.6/2.5	—	No	Yes (strong)
pxtemp	rain/snow threshold temperature	−1–3 °C	—	No	Yes
mfmax	max melt factor (Jun 21)	0.5–2.5	1.2	Yes	Weak
mfmin	min melt factor (Dec 21)	0.05–0.5	0.20	Yes	Weak
tipm	antecedent temperature index weight	0.1–1.0	0.20	Yes	Weak
nmf	max negative (refreeze) melt factor	0.05–0.5	0.15	Yes	Weak
plwhc	liquid-water holding capacity	0.02–0.3	0.04	Yes	Weak
uadj	wind / rain-on-snow melt factor	0.03–0.19	0.08	Yes	Very weak
mbase	base melt temperature	fixed 0	0.0	fixed	—

Technical setup (as run)

Inputs / outputs

Truth	SNODAS SWE (NSIDC G02158), ~8-daily, aggregated to 0.1°
Forcing	ERA5-Land daily T2m + total precip; per-cell temperature correction from SNOTEL bias
Elevation	SRTM aggregated to 0.1° (atmospheric pressure term)
Output	9 parameter maps + test-KGE + warming + masks → snow17_conus_era5.nc (250×585)

Domain / time / training

Grid	CONUS 0.1° · 585×250 · 58,265 snow cells
Period	WY2010-10-01 → 2024-09-30 (15 yr, 5,114 daily steps)
Train / test	temporal holdout — first 66% train, last 34% test
Optimizer	Adam + BPTT, two-stage (regional → per-cell), spatial reg λ=2, physical prior λ=200
Compute	tiled 2.5° · 46 parallel CPU workers · ~2.5 h

Validation is threefold: temporal SNODAS holdout (reported above), snowpack-stratified skill (peak≥100 mm), and independent SNOTEL ground stations held out of the loss.

Limitations & next

Known limits

Maritime deep snow (Sierra/Cascades) under-simulates — sub-grid orography + rain/snow partition, not a tuning gap.
Melt-out timing vs MODIS fSCA disagrees by ~2 weeks — a real SNODAS/fSCA tension, deferred (needs denser fSCA + more years).
Eastern temperature correction is zero (no SNOTEL east); GHCN/COOP would extend it.
Prior targets are empirical, not physical law — appropriate only because the forcing is bias-free.

What's next

Stage 2 — water balance: port the CREST runoff kernel to a differentiable form, verified against the EF5 C++, then calibrate to basin runoff.
Optional: PRISM orographic precip for maritime; GHCN east correction.
Global roll-out using the regional-stage parameters + prior as warm-start.