CREST-Global · Stage-2 Water Balance

Headline

Runoff is a model flux — never observed directly. We bridge it to observed streamflow at monthly scale (routing delay washes out), and learn a small, physical multiplier field.

The problem

Water balance produces per-grid runoff; gauges measure streamflow (routed + integrated). There is no gridded runoff truth. We aggregate grid runoff over each basin and match monthly volume — routing timing averages out.

The parameterization

θ = multiplier × EF5 a-priori grid (the NWS/FLASH scheme). We optimize a handful of per-basin multipliers, not free per-cell params — the a-priori supplies spatial structure, killing equifinality.

The optimization loop

Six structured experiments, each selected on a strict temporal holdout: baseflow reservoir, interflow split, daily Oudin PET, quickflow smoothing, bound repairs. Holdout KGE 0.56 → 0.67; the arid West rescued from −0.11 to +0.41.

Results — after a six-experiment optimization loop

Per-basin KGE on monthly runoff depth (mm/month), 424 basins. Every design choice was selected on a strict temporal holdout (train WY2011–2020, test WY2020–2024) — never on the training score.

Step	change	holdout TEST median KGE
baseline	CREST + groundwater baseflow reservoir (gwc)	0.56
E1	learnable interflow split φ (→GW vs direct)	0.59
E2	daily Oudin PET from ERA5 T (interannual ET) replaces monthly climatology	0.62
E3	quick reservoir on fastflow + full convergence	0.64
E5–E6	bounds fixed where multipliers railed (ke, gwc, b)	0.67
Round 2 — residual autopsy (seasonal-bias profiles, worst-basin forensics) generated five more physically-motivated hypotheses, each tested against a measured optimization-noise floor of ±0.002:
r1	channel / deep loss fraction (arid basins over-produced ×5–13)	0.69
r3	fastflow → groundwater recharge (West: spring-melt water must return in fall)	0.69
r4	per-basin liquid-input multiplier (residual forcing bias — biggest single lever)	0.70
r2, r6, r7, E7	seasonal ET, FEWS/hybrid PET, log-loss — tested and rejected	≤ 0.67
c1 combo	all three winners together	0.72

Final model (c1 physics)	median KGE	KGE > 0.5	KGE > 0
Temporal-holdout TEST (unseen WY2020–2024)	0.72	74%	91%
Full-record production calibration	0.79	80%	—
GAGES-II Reference (near-natural) subset, TEST	0.74*	79%	—

*E6-configuration number; only 34 of our 424 basins are GAGES-II Reference — most residual error sits in human-influenced basins (Non-ref TEST median 0.66), not in the model physics.

Two findings from the loop: (1) interannual PET matters — replacing climatological PET with daily Oudin PET jumped timing r from 0.59 to 0.78; (2) the arid West's failure in the dry test years was a bound artifact (ke railed, model couldn't evaporate enough) — widening it rescued the West from median KGE −0.11 to +0.41. Snow-dominated basins (44% of the set) perform as well as snow-free ones (0.66 vs 0.70) — the frozen Stage-1 snowmelt coupling holds up.

Stage-2 KGE map, distribution, multipliers, example hydrographs — Final production model (c1 physics, full record): (A) per-basin KGE over CONUS (median 0.79); (B) KGE CDF; (C) calibrated multiplier distributions; (D–G) example monthly runoff, observed (USGS) vs simulated, across four regions.

Where the parameters and skill sit in space

Spatial maps of calibrated multipliers and KGE components — Top: calibrated multipliers (× the EF5 a-priori; 1.0 = a-priori) and the two reservoir constants. Bottom: KGE and its components r (timing), β (bias), α (variability). The West needs more soil storage (WM) and more ET (KE); skill is high in the East, lower in the arid West.

Generalization — an honest holdout

All numbers above marked TEST come from parameters that never saw WY2020–2024. Sample sizes, stated plainly:

Temporal holdout (final model)	median KGE	KGE > 0.5	KGE > 0
TRAIN — calibrated on WY2011–2020 (102 months)	0.76	81%	—
TEST — held-out WY2020–2024 (54 months)	0.67	70%	90%

The train→test drop is −0.09 — small, because each basin has only 7 low-DOF multipliers plus a prior toward the a-priori. Note on sampling: a single basin's test KGE rests on 54 monthly pairs (±~0.1 noise); the reported medians rest on 424 basins × 54 ≈ 23,000 basin-months, so population statistics and experiment rankings are robust even where individual basins are noisy.

Spatial generalization — dPL to ungauged basins

A shared MLP g(64-dim AlphaEarth satellite embedding, basin mean) → the ten multipliers, trained end-to-end through the differentiable water balance on the 424 basins' monthly runoff — then applied to basins the model has never seen. A 5-member seed ensemble (multipliers averaged) is the production regionalizer.

Ungauged-prediction test	median KGE	KGE > 0.5	KGE > 0
PUB — leave-basins-out CV (212 held-out basins, ensemble)	0.65	63%	90%
Independent judge — 295 GAGES-II Reference basins, never used in any calibration, zero-shot	0.52	52%	86%
vs nearest-basin multiplier transfer	0.45	44%	74%
vs a-priori only (uncalibrated)	−1.31	1%	11%

Read: an ungauged CONUS basin gets ~80% of gauged-calibration skill via dPL (0.65 vs 0.79), and on a fully independent near-natural test set dPL beats geographic nearest-transfer decisively in the tail (86% vs 74% positive). Findings from the dPL loop: the AlphaEarth embedding alone suffices (adding hand-crafted covariates or capacity did not help); seed-ensembling is the one lever that reliably lifts skill (+0.05 PUB). The v2 parameter product uses per-basin calibration inside gauged basins and the dPL ensemble everywhere else — this is also the template for the global rollout (same embedding space worldwide).

Full-coverage v2 water-balance parameter fields — v2 full-coverage parameter fields at 0.1° — per-basin calibration inside the 424 gauged basins, dPL-ensemble multipliers from basin-scale-mean (0.7°) AlphaEarth embeddings everywhere else, clipped to the calibrated multiplier range. The arid-West patterns (high loss fraction lf, high recharge fgw) are hydrologically plausible for ephemeral streams but sit in the least-constrained part of the product — ungauged desert cells differ from the gauged mountain basins embedded within them.

Methods

1 · Data

Truth	USGS daily streamflow (waterservices REST) → monthly volume → depth (mm/mo), 424 basins
Basins	USGS NLDI upstream polygons rasterized to 0.1°; filtered by area agreement, ≥5 yr of months, runoff-ratio (drops regulated/diverted)
A-priori params	EF5-US-Parameters (NWS/FLASH 2016): WM, B, IM, ksat(=FC) at 1 km → aggregated to 0.1°
PET	daily Oudin PET from ERA5-Land temperature (0.408·Ra·(T+5)/100) — interannual variability; the EF5 monthly climatology was tested and retired (E2: r 0.59→0.78)
Liquid input	frozen Stage-1 Snow-17 outflow (rain-through + snowmelt), computed once

2 · Yes — snowmelt is the input

The water-balance forcing is not raw precipitation but the liquid water leaving the snowpack from the frozen Stage-1 Snow-17: rain passes straight through in the warm season, snowmelt drives the spring pulse. This is what couples the two stages.

3 · Model & parameterization

θ_k(cell) = m_k^basin × apriori_k(cell) for k ∈ {WM, B, IM}; KE = m_KE; FC fixed

Per cell per day: CREST → fastflow (overland) + slowflow (interflow).
Input scaling: liquid input × pm (round-2: residual forcing-bias absorber).
GW recharge: a fraction fgw of fastflow enters the groundwater store before routing (round-2: spring-melt → fall-baseflow pathway).
Quick reservoir: remaining fastflow through a linear store released at qc/day (≈0.7 → 1–2-day smoothing).
Baseflow reservoir: a learnable fraction φ of interflow (≈1.0 in practice) fills a linear groundwater store released at gwc/day → baseflow. (Concept & name from the CRESTPhys EF5 fork.)
Channel / deep loss: (1 − lf) of the total reaches the gauge (round-2: arid transmission loss).
FC is fixed — without routing it only controls the fast/slow split, which cancels in the monthly total (unidentifiable until Stage-3).
Grid runoff summed per month, area-averaged over each basin → simulated monthly depth.

4 · Loss & metric

KGE = 1 − √[(r−1)² + (α−1)² + (β−1)²] · L = mean_basin[ 1 − ½(KGE + KGE√) + 5(β−1)² ] + λ·(m → a-priori)

A dual-transform KGE (raw + √) balances high and low flows; the explicit β term hard-anchors long-term water balance; a mild prior keeps multipliers near the a-priori. Optimized with Adam on GPU; 5114-day BPTT memory bounded by per-month gradient checkpointing.

Calibrated multipliers

Median multiplier over the 424 basins (1.0 = trust the EF5 a-priori exactly).

Parameter	controls	median	note
WM ×	max soil water capacity	1.06	≈ a-priori once pm carries the water-amount correction
B ×	infiltration-curve shape	0.92	near a-priori
IM ×	impervious fraction	0.96	≈ 1 (NLCD a-priori accurate)
KE ×	PET scaling	1.48	effective — compensates Oudin PET's low bias (down from 2.35 once pm/lf exist)
pm	liquid-input multiplier	0.79	ERA5 rain+melt runs systematically high vs gauged runoff; many basins near the 0.7 floor
lf	channel / deep loss fraction	0.06	largest in arid basins (transmission loss)
fgw	fastflow → GW recharge	0.11	spring melt stored, released as fall baseflow
gwc	groundwater release /day	0.051	~20-day recession; slower in the West
φ	interflow → GW fraction	0.99	nearly all interflow routes through the GW store
qc	quickflow release /day	0.70	1–2-day overland smoothing

Multipliers are strictly positive (sigmoid-bounded) and anchored to the EF5 a-priori by a mild prior — the a-priori supplies the spatial pattern, calibration applies regional corrections. Adding the physically-targeted round-2 parameters (pm, lf, fgw) relieved the older compensations: WM returned from 1.57 to 1.06 and KE from 2.35 to 1.48 — each new process variable made the remaining parameters more physical. pm clustering near its 0.7 floor flags a systematic wet bias in the ERA5-driven liquid input as the next forcing question.

Limitations & next

Known limits

Arid/snow West is the hard region — timing (r) and flashiness (α) drag KGE there, mirroring Stage-1's western difficulty.
Spatial (ungauged) generalization not yet tested — only temporal holdout so far.
Baseflow reservoir lives in the calibration layer — should be folded into the CREST kernel (CRESTPhys-style) with an aquifer cap.
PET is a monthly climatology (no interannual ET variability).

What's next

dPL regionalization: attributes → multipliers, to predict ungauged basins & produce a continuous 0.1° field.
Stage-3 routing on MERIT-Hydro / EF5 flow grids — makes FC identifiable, moves to daily hydrographs.
Package pyCREST (kernel + baseflow + training harness).