Unified notation

One reference list for every symbol used in the HydroModPy theory pages. SI units are the default. Per-page math notes link back here via the labels at the end of each row, so the same variable is never defined twice.

Hydraulic state and forcing

Symbol	Description	SI unit	Notes / cross-refs
h	Hydraulic head measured from a fixed datum	m	Field variable solved by every flow backend; see MODFLOW Governing Equation And CVFD Formulation.
z_top	Land surface elevation	m	Used in the Boussinesq surface-interaction closures.
z_bot	Aquifer bottom elevation	m	Domain depth declared via [domain.depth_model].
b = h - z_bot	Saturated thickness	m	Drives transmissivity in unconfined regimes.
q_x, q_y, q_z	Specific discharge along axis	m s-1	Darcy flux per unit area; see Solver Architecture.
Q	Volumetric discharge	m3 s-1	Streamflow, well rate, drainage outflow.
R	Recharge rate (positive into the aquifer)	m s-1	Forced via [flow.bc] recharge entries.
q_ex	Surface-interaction exchange flux	m s-1	Boussinesq closure between aquifer and ground surface.

Hydraulic parameters

Symbol	Description	SI unit	Notes / cross-refs
K	Saturated hydraulic conductivity	m s-1	Field declared in [flow.param.K]; supports homogeneous, heterogeneous, anisotropic.
T = K * b	Transmissivity	m2 s-1	Used in linearized Boussinesq variants.
S_s	Specific storage	m-1	Storage per unit volume.
S_y	Specific yield	dimensionless	Drainable porosity in unconfined regimes.
n	Porosity	dimensionless	Required for transport formulations.
alpha_L, alpha_T	Longitudinal and transverse dispersivities	m	Transport package input (MODFLOW 6 GWT, MT3DMS).

Geometry and discretisation

Symbol	Description	SI unit	Notes / cross-refs
L	Domain characteristic length (1D, 2D)	m	Used in analytical validation cases.
A	Catchment surface area	m2	Reported per project; gallery shows area_km2.
dx, dy	Cell size on a structured grid	m	Set via [<solver>.sgrid.planar].
nlay	Number of vertical layers	dimensionless	Set via [<solver>.sgrid.vertical].
dt	Solver internal timestep	s	Adaptive in PETSc backend; fixed in MODFLOW.
Pe	Mesh Peclet number for transport	dimensionless	Used to assess oscillations on coarse meshes.

Time and calibration

Symbol	Description	SI unit	Notes / cross-refs
t	Simulation time	s	Reported in catalog as ISO date when calendar metadata exists.
T_sim	Total simulated horizon	s	Set via [simulation.time].
RMSE	Root-mean-square error of an observable	same as observable	Default calibration objective.
NSE	Nash-Sutcliffe efficiency	dimensionless	Available via [calibration.objective].
KGE	Kling-Gupta efficiency	dimensionless	Optional alternative to NSE.
NSElog	Nash-Sutcliffe efficiency on log-transformed series	dimensionless	Weights the recessions rather than the peaks. Not to be confused with transform = "log", which takes the logarithm of an already-computed cost; the two names look alike and mean unrelated things.
D_so	Mean downslope distance from the simulated stream network to the mapped one	m	A tail statistic, not a typical gap: its median is usually zero. See Downslope-distance calibration of the stream network.
D_os	Mean downslope distance from the mapped stream network to the simulated one	m	Same page. D_so large means an excess of simulated stream, D_os large a deficit.
J	|D_so - D_os|, the calibrated cost	m	Its zero is the intersection of two curves, not the minimum of a distance. The signed residual travels separately, because a root search needs its sign.
J_signed	D_so - D_os, the residual the root search drives to zero	m	Positive means an excess of simulated stream, negative a deficit. It travels in the trial components and never in the cost, so picking the best trial by lowest cost gives the one closest to zero rather than the most negative one.
Doptim	(D_so + D_os) / 2	m	A diagnostic. Legitimate as a cost only in the outer loop that picks between structures already balanced at J = 0.
roptim	Doptim / L_ref	dimensionless	Validity indicator. It measures agreement, never correctness: measured on synthetic truth it improves as the bias worsens, and it improved from 4.58 to 2.21 on one catchment purely because the grid was coarser. Two values from different meshes are not comparable.
L_ref	Square root of the median cell area over the catchment	m	The median and not the mean: a few large buffer cells would otherwise set the scale. Declaring observed_position_accuracy floors it, so that refining the mesh stops shrinking the denominator on its own.
L_cap	Longest downslope descent to the outlet inside the catchment	m	The value a cell whose descent never meets its target saturates at. Counting it as +inf makes the mean infinite; dropping it moves the support with the calibrated parameter.
alpha	Ratio of the mapped network to its own downslope closure	dimensionless	Measures the agreement between the routing surface and the mapped network. Below 0.90, the distances read a dataset disagreement rather than hydrogeology.
T/R	Transmissivity over recharge	m	The output to publish. K inherits the recharge series entirely, measured at plus or minus 25 per cent between reanalyses.

Conventions

• All times in TOML accept SI units ("3600 s", "1 h", "0.5 d"). The Pydantic Pint integration normalises them.

• Lengths default to metres; declarations may override with explicit units ("500 m", "2 km").

• Vector quantities follow the right-hand rule with z pointing up.

• Sign convention: Q > 0 and R > 0 represent flux into the aquifer.

For the in-scope backends and what each one represents, see Solver Capability Matrix.