Symbols and notation#

This page collects the main mathematical symbols, data symbols, and diagnostic notation used throughout GeoPrior-v3.

Sign conventions#

Important

GeoPrior uses both depth-like and head-like quantities.

\(d(i,t)\) is groundwater depth below ground surface and is positive downward.
\(h(i,t)\) is hydraulic head and is positive upward.
Drawdown is typically written as \(\Delta h = h_{\mathrm{ref}} - h\), so positive \(\Delta h\) corresponds to head loss.

Data and workflow symbols#

Symbol	Units	Meaning
\(s(i,t)\)	m	Cumulative vertical subsidence at grid cell \(i\) and year \(t\).
\(d(i,t)\)	m	Groundwater depth below ground surface, positive downward.
\(h(i,t)\)	m	Hydraulic head, positive upward.
\(R(i,t)\)	mm yr^-1	Annual rainfall.
\(U^*(i,t)\)	dimensionless	Normalised shallow urban-load proxy. This is a relative geospatial loading indicator, not a calibrated stress field.
\(H(i)\)	m	Mapped geological thickness in the harmonised data archive.
\(H_{\mathrm{eff}}(i)\) or \(H_{\mathrm{eff}}(x,y)\)	m	Censor-aware effective compressible thickness. This is supplied by the data pipeline and is not learned as a free inversion field.
\(z_{\mathrm{surf}}(i)\)	m	Surface-elevation datum used in head construction and related preprocessing.
\(c_H(i)\)	dimensionless	Right-censor indicator for the thickness field.

Physics and closure symbols#

(1)#\[\tau \approx \frac{H_d^2 \, S_s}{\pi^2 \, \kappa_b \, K}\]

The reduced closure above is central to GeoPrior and links the effective drainage path, storage, anisotropy/leakage factor, and hydraulic conductivity to the learned consolidation timescale.

Symbol	Units	Meaning
\(K(x,y)\)	m s^-1	Effective horizontal hydraulic conductivity controlling lateral head diffusion and entering the timescale closure.
\(S_s(x,y)\)	m^-1	Specific storage. Links head changes to volumetric strain and enters both groundwater balance and equilibrium settlement.
\(H_d(x,y)\)	m	Effective drainage thickness or drainage path length used in the consolidation-timescale closure.
\(\tau(x,y)\)	s	Learned consolidation or relaxation timescale.
\(\tau_{\mathrm{closure}}\)	s	Closure-implied consolidation timescale computed from \(K\), \(S_s\), and \(H_d\).
\(m_v(x,y)\)	Pa^-1	Vertical compressibility. Relates strain to effective stress and supports the approximation \(S_s \approx m_v \gamma_w\).
\(\gamma_w\)	N m^-3	Unit weight of water, used to convert head change to effective-stress change.
\(k_v\)	m s^-1	Effective vertical drainage conductivity.
\(\kappa_b\)	dimensionless	Effective anisotropy or leakage factor linking vertical conductivity to horizontal conductivity through \(k_v = \kappa_b K\).
\(C_v\)	m² s^-1	Consolidation coefficient.
\(\Delta h\)	m	Drawdown, typically written as \(\Delta h = h_{\mathrm{ref}} - h\).
\(h_{\mathrm{ref}}\)	m	Reference head datum used to define drawdown.
\(s_{\mathrm{eq}}\)	m	Equilibrium settlement implied by the current head state.
\(H_{\mathrm{phys}}\)	m	Physical compressible-layer thickness appearing in the 1-D consolidation derivation.
\(w(z,t)\)	m	Vertical displacement in the 1-D consolidation derivation, taken positive downward.
\(\varepsilon_v(z,t)\)	dimensionless	Vertical strain, commonly written as \(\varepsilon_v = \partial w / \partial z\).
\(\Delta \sigma'\)	Pa	Effective-stress change induced by drawdown.
\(Q(t,x,y)\)	context-dependent	Effective forcing term appearing in the groundwater-flow residual. In the code and docs it is treated as the forcing contribution in the reduced groundwater equation.

Residuals and diagnostic symbols#

(2)#\[R_{\mathrm{gw}} = S_s \, \partial_t h - \nabla \cdot \left(K \nabla h\right) - Q\]

(3)#\[R_{\mathrm{cons}} = \partial_t s - \frac{s_{\mathrm{eq}}(h) - s}{\tau}\]

These are the two main reduced residuals enforced by the GeoPrior physics head.

Symbol	Units	Meaning
\(R_{\mathrm{gw}}\)	reduced residual	Groundwater-flow residual.
\(R_{\mathrm{cons}}\)	reduced residual	Consolidation residual in relaxation form.
\(R_{\mathrm{prior}}\)	log-residual	Prior-consistency residual comparing learned \(\tau\) with closure-implied \(\tau\).
\(\varepsilon_{\mathrm{cons}}\)	dimensionless	RMS dynamic-balance or consolidation-error diagnostic.
\(\varepsilon_{\mathrm{prior}}\)	dimensionless	Prior-consistency error diagnostic for timescale mismatch.
\(\Delta \log_{10}\tau\)	decades	Closure-tension or identifiability-style mismatch in log-timescale space.

Forecasting and probabilistic symbols#

Symbol	Units	Meaning
\(L\)	years	Look-back window length.
\(H\)	years	Forecast horizon length.
\(T_{\mathrm{eval}}\)	time steps	Number of time steps in the evaluation window.
\(Q\)	set of probabilities	Forecast quantile set. In the main manuscript setting, \(Q = \{0.1, 0.5, 0.9\}\).
\(\hat{s}^{(q)}(i,t_h)\)	m	Predicted subsidence at quantile \(q\).
\(\hat{h}^{(q)}(i,t_h)\)	m	Predicted hydraulic head at quantile \(q\).
\(PI_{0.8}(i,t_h)\)	m	Nominal central 80% prediction interval, typically \([\hat{s}^{(0.1)}, \hat{s}^{(0.9)}]\).
\(T_s\)	mm yr^-1	Annual subsidence threshold used in exceedance tests.
\(m_{i,t_h}\)	binary mask	Availability mask for future head supervision when only a subset of windows has groundwater-head targets.

Model-dimension symbols#

Symbol	Units	Meaning
\(d\)	latent width	Stream embedding width in the predictive backbone.
\(d_f\)	latent width	Fused latent width after feature fusion.
\(d_d\)	latent width	Decoder width.
\(n_{\mathrm{LSTM}}\)	count	Number of LSTM layers.
\(h_{\mathrm{LSTM}}\)	count	Number of LSTM units per layer.
\(n_{\mathrm{head}}\)	count	Number of attention heads.
\(p_{\mathrm{drop}}\)	probability	Dropout rate.
\(\eta\)	learning rate	Optimizer learning rate.
\(B\)	batch size	Mini-batch size.
\(N_{\max}\)	epochs	Maximum number of training epochs.

Notes on interpretation#

\(H_{\mathrm{eff}}\) is externally supplied and should be read as an effective, censor-aware thickness representation rather than as an uncensored borehole-thickness estimate.
\(U^*\) is a dimensionless proxy for shallow areal loading potential and should not be interpreted as a stress field in Pa.
The most interpretable effective outputs are usually the relative spatial patterns, closure-consistent timescale structure, and thickness pathway, rather than a claim of unique local parameter recovery.