Equations and implementations
This package contains named model variants. The equations below describe the implemented variants and make the time step, units, initial states and routing choices explicit. They are not blanket claims of equivalence to every implementation carrying the same model name. Source links point to the vendored equations. P, E, Q and stores are water depth in millimetres per model step unless a routed volume or discharge is stated.
GR4J daily kernel (LuMod 0.1.3.0)
States are production store S (capacity x1 mm), routing store R (capacity x3 mm), and the two unit-hydrograph queues. Initial stores are fractions ps0*x1 and rs0*x3; queues start empty. Let Pn=max(P-E,0), En=max(E-P,0). For P>E,
For E>=P,
Then S*=S-Es+Ps; percolation is Perc=S* - S*/(1+(S*/(2.25 x1))^4)^(1/4), with S=S*-Perc and P_r += Perc. The x4 unit hydrographs use ordinates that are successive differences of F1(t)=min(max(t/x4,0),1)^2.5 and F2(t)=0.5(t/x4)^2.5 up to x4, then 1-0.5(2-t/x4)^2.5 up to 2x4, then one. 90% of routed input enters UH1, 10% UH2. Groundwater exchange is F=x2(R/x3)^3.5; routing store is updated by UH1 plus F, clipped at zero, then nonlinear routing release uses exponent four. Direct release is max(0,0.1*UH2+F); total is direct plus routing-store release. Pinned equation kernel.
GR2M monthly kernel (LuMod 0.1.3.0)
Initial production storage is s0*x1 mm and routing storage r0*60 mm. With phi=tanh(P/x1) and psi=tanh(E/x1), production store transforms as S1=(S+x1 phi)/(1+phi S/x1), P1=P+S-S1, S2=S1(1-psi)/(1+psi(1-S1/x1)), then capacity correction S=S2/(1+(S2/x1)^3)^(1/3). The effective input is P1+S2-S. Add this to routing storage R1; apply exchange/scaling R2=x2(R+P1+S2-S). Monthly runoff is Q=R2^2/(R2+60) and the next routing state is R=R2-Q. Pinned kernel.
GR1A annual kernel (LuMod 0.1.3.0)
For the first year u=P/(x E). Thereafter u=(0.7 P_t+0.3 P_{t-1})/(x E_t). Annual runoff depth is Q=P_t[1-(1+u^2)^(-1/2)]. This implementation uses the immediately preceding annual precipitation and requires positive annual PET. Pinned kernel.
HYMOD2 daily kernel (LuMod variant)
This is the source implementation of HYMOD2, not the classic five-parameter model. Set Cmax=Wmax/(1+b), b=log2(1/(1-beta/2)), and C(W)=Cmax(W/Wmax)^(1+b). Given previous soil water W, first calculate overflow Peff2=max(0,P+W-Wmax), infiltration I=P-Peff2, and interim water W*=min(Wmax,I+W). Then Peff1=max(0,I+C(W)-C(W*)), total effective rain Peff=Peff1+Peff2. The ET coefficient is k=kmin+(kmax-kmin)(C(W*)/Cmax)^ce, with kmin=llet*kmax, ce=log2(1/(1-cexp/2)); ET is k*min(PET,C(W*)). Soil content after ET is inverted through the capacity curve to update W. Effective rain partitions into quick alpha*Peff and slow (1-alpha)*Peff. Slow and each of nres serial quick stores are linear reservoirs (out=k*store; next store max(0,store-out+inflow)). Initial soil storage is w0*Wmax; quick and slow stores use wq0 and ws0. Pinned source equations.
HBV-light snow daily variant
Temperature-index snow uses M=min(Snow,dd(T-Tt)) for T>=Tt and M=0 otherwise; below threshold all precipitation becomes snow, otherwise liquid input is P+M. Effective precipitation is Pliq(SM/FC)^beta, with remaining liquid infiltrating. The pinned source passes the initial parameter s0 into its ET routine at each step: ET=PET if s0>pwp, otherwise ET=PET*s0/(pwp*FC). This follows the implementation even though it does not use the evolving SM state for ET. The upper store receives runoff and successively releases threshold flow k0*max(S1-lthres,0), interflow k1*S1, and percolation kp*S1, each limited by available upper-store water. The lower store receives percolation and releases k2*S2. Total runoff is routed with a triangular unit hydrograph governed by maxbas. Initial snow, soil fraction and upper/lower stores are snow0, s0*FC, w01 and w02; routing queues start empty. Pinned kernel.
MILC daily variant
The single-layer soil water balance uses a capacity-dependent infiltration fraction (1-(W/Wmax)^alpha), actual ET max(0,PET*W/Wmax), and power-law drainage KS*(W/Wmax)^m, partitioned into baseflow (1-nu) and percolation nu. Excess rainfall is runoff. The package supplies explicit PET, so the upstream PET submodel is not called. The upstream GIUH routing uses its vendored tabulated curve and dt=0.2; the lookup table and gamma/Nash convolution are part of the pinned kernel. Initial soil fraction is w0; channel-routing queues initialize empty. Pinned kernel and table.
GR3J, GR5J and GR6J
These are the named equations at the pinned hydromodel revision. The model family uses a production store with nonlinear rainfall/evaporation transformations, S-curve unit hydrograph routing, exchange terms and variant-specific additional stores/fluxes. For GR3J, production capacity is fixed at 330 mm; initial production and routing stores are set by the pinned source’s sinusoidal initial-state formulas; the routing exchange is F=x2(R/x2)^4, and runoff is the sum of the two routed components. GR5J and GR6J add variant-specific states, exchange or evapotranspiration terms. Full terms and parameter ordering are kept in the cited executable source rather than replaced with textbook-family shorthand: GR3J, GR5J, GR6J. Initial states and helper equations in that implementation are part of the model contract.
Classic HYMOD and XAJ
HYMOD_CLASSIC is the separate five-parameter hydromodel implementation: its complete soil curve, quick/slow partition, serial reservoirs, and initial-state defaults are given in pinned source. XAJ and XAJ_MZ implement the source’s three-layer tension-water accounting, runoff generation, free-water partitioning and channel routing; their full flux/state equations, parameter names/order and unit hydrograph are in XAJ source. XAJ_MZ changes source equations and parameterization; it is not a cosmetic name variant.
SMART daily wrapper
Each daily P and PET depth is apportioned uniformly to 24 hourly calls to the pinned SMARTpy catchment and river step equations. The catchment returns hourly runoff-component discharges; their sum enters a linear river store with Qriver=Sriver/(RK*3600), and Sriver is advanced by hourly inflow volume less outflow volume, constrained nonnegative using the upstream 95% availability rule. Soil layers initialize equally at Z/12 mm; runoff-routing stores initialize empty. Daily simulated depth is the sum of hourly river volumes divided by catchment area. This daily-to-hourly disaggregation is a forcing assumption. Catchment and river equations.
ABCD monthly water balance
With previous soil store S and precipitation P, available water is W=S+P. The rationalized root is algebraically identical to the literal root: Y=2bW/[W+b+sqrt((W+b)^2-4abW)]. Soil after ET is S'=Y exp(-PET/b), so AET=Y-S'. Surplus R=W-Y partitions to groundwater recharge cR and quick flow (1-c)R; groundwater updates to G'=(G+cR)/(1+d), and streamflow is Q=(1-c)R+dG'. Default initial stores are zero and the model is monthly. Implementation; equation reference.
MARRMoT structures
The optional MARRMoT implementations follow the exact vendored .m model_fun state derivatives, source-ordered p01... values, flux helper equations, and implicit-Euler solver. s01... are explicit fixed stores (zero by default). These 47 distinct structures do not share one lumped-model equation. See the matching generated model page for source, frequency, calibrated parameter ranges and stores; use its exact upstream model_fun and called flux helper as the governing equations. They currently execute through Octave and are not yet native Python kernels.