.. _symbols: 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. - :math:`d(i,t)` is **groundwater depth below ground surface** and is positive **downward**. - :math:`h(i,t)` is **hydraulic head** and is positive **upward**. - Drawdown is typically written as :math:`\Delta h = h_{\mathrm{ref}} - h`, so positive :math:`\Delta h` corresponds to head loss. Data and workflow symbols ------------------------- .. list-table:: :header-rows: 1 :widths: 18 16 66 * - Symbol - Units - Meaning * - :math:`s(i,t)` - m - Cumulative vertical subsidence at grid cell :math:`i` and year :math:`t`. * - :math:`d(i,t)` - m - Groundwater depth below ground surface, positive downward. * - :math:`h(i,t)` - m - Hydraulic head, positive upward. * - :math:`R(i,t)` - mm yr\ :sup:`-1` - Annual rainfall. * - :math:`U^*(i,t)` - dimensionless - Normalised shallow urban-load proxy. This is a relative geospatial loading indicator, not a calibrated stress field. * - :math:`H(i)` - m - Mapped geological thickness in the harmonised data archive. * - :math:`H_{\mathrm{eff}}(i)` or :math:`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. * - :math:`z_{\mathrm{surf}}(i)` - m - Surface-elevation datum used in head construction and related preprocessing. * - :math:`c_H(i)` - dimensionless - Right-censor indicator for the thickness field. Physics and closure symbols --------------------------- .. math:: \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. .. list-table:: :header-rows: 1 :widths: 20 16 64 * - Symbol - Units - Meaning * - :math:`K(x,y)` - m s\ :sup:`-1` - Effective horizontal hydraulic conductivity controlling lateral head diffusion and entering the timescale closure. * - :math:`S_s(x,y)` - m\ :sup:`-1` - Specific storage. Links head changes to volumetric strain and enters both groundwater balance and equilibrium settlement. * - :math:`H_d(x,y)` - m - Effective drainage thickness or drainage path length used in the consolidation-timescale closure. * - :math:`\tau(x,y)` - s - Learned consolidation or relaxation timescale. * - :math:`\tau_{\mathrm{closure}}` - s - Closure-implied consolidation timescale computed from :math:`K`, :math:`S_s`, and :math:`H_d`. * - :math:`m_v(x,y)` - Pa\ :sup:`-1` - Vertical compressibility. Relates strain to effective stress and supports the approximation :math:`S_s \approx m_v \gamma_w`. * - :math:`\gamma_w` - N m\ :sup:`-3` - Unit weight of water, used to convert head change to effective-stress change. * - :math:`k_v` - m s\ :sup:`-1` - Effective vertical drainage conductivity. * - :math:`\kappa_b` - dimensionless - Effective anisotropy or leakage factor linking vertical conductivity to horizontal conductivity through :math:`k_v = \kappa_b K`. * - :math:`C_v` - m\ :sup:`2` s\ :sup:`-1` - Consolidation coefficient. * - :math:`\Delta h` - m - Drawdown, typically written as :math:`\Delta h = h_{\mathrm{ref}} - h`. * - :math:`h_{\mathrm{ref}}` - m - Reference head datum used to define drawdown. * - :math:`s_{\mathrm{eq}}` - m - Equilibrium settlement implied by the current head state. * - :math:`H_{\mathrm{phys}}` - m - Physical compressible-layer thickness appearing in the 1-D consolidation derivation. * - :math:`w(z,t)` - m - Vertical displacement in the 1-D consolidation derivation, taken positive downward. * - :math:`\varepsilon_v(z,t)` - dimensionless - Vertical strain, commonly written as :math:`\varepsilon_v = \partial w / \partial z`. * - :math:`\Delta \sigma'` - Pa - Effective-stress change induced by drawdown. * - :math:`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 -------------------------------- .. math:: R_{\mathrm{gw}} = S_s \, \partial_t h - \nabla \cdot \left(K \nabla h\right) - Q .. math:: 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. .. list-table:: :header-rows: 1 :widths: 24 16 60 * - Symbol - Units - Meaning * - :math:`R_{\mathrm{gw}}` - reduced residual - Groundwater-flow residual. * - :math:`R_{\mathrm{cons}}` - reduced residual - Consolidation residual in relaxation form. * - :math:`R_{\mathrm{prior}}` - log-residual - Prior-consistency residual comparing learned :math:`\tau` with closure-implied :math:`\tau`. * - :math:`\varepsilon_{\mathrm{cons}}` - dimensionless - RMS dynamic-balance or consolidation-error diagnostic. * - :math:`\varepsilon_{\mathrm{prior}}` - dimensionless - Prior-consistency error diagnostic for timescale mismatch. * - :math:`\Delta \log_{10}\tau` - decades - Closure-tension or identifiability-style mismatch in log-timescale space. Forecasting and probabilistic symbols ------------------------------------- .. list-table:: :header-rows: 1 :widths: 18 16 66 * - Symbol - Units - Meaning * - :math:`L` - years - Look-back window length. * - :math:`H` - years - Forecast horizon length. * - :math:`T_{\mathrm{eval}}` - time steps - Number of time steps in the evaluation window. * - :math:`Q` - set of probabilities - Forecast quantile set. In the main manuscript setting, :math:`Q = \{0.1, 0.5, 0.9\}`. * - :math:`\hat{s}^{(q)}(i,t_h)` - m - Predicted subsidence at quantile :math:`q`. * - :math:`\hat{h}^{(q)}(i,t_h)` - m - Predicted hydraulic head at quantile :math:`q`. * - :math:`PI_{0.8}(i,t_h)` - m - Nominal central 80% prediction interval, typically :math:`[\hat{s}^{(0.1)}, \hat{s}^{(0.9)}]`. * - :math:`T_s` - mm yr\ :sup:`-1` - Annual subsidence threshold used in exceedance tests. * - :math:`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 ----------------------- .. list-table:: :header-rows: 1 :widths: 18 16 66 * - Symbol - Units - Meaning * - :math:`d` - latent width - Stream embedding width in the predictive backbone. * - :math:`d_f` - latent width - Fused latent width after feature fusion. * - :math:`d_d` - latent width - Decoder width. * - :math:`n_{\mathrm{LSTM}}` - count - Number of LSTM layers. * - :math:`h_{\mathrm{LSTM}}` - count - Number of LSTM units per layer. * - :math:`n_{\mathrm{head}}` - count - Number of attention heads. * - :math:`p_{\mathrm{drop}}` - probability - Dropout rate. * - :math:`\eta` - learning rate - Optimizer learning rate. * - :math:`B` - batch size - Mini-batch size. * - :math:`N_{\max}` - epochs - Maximum number of training epochs. Notes on interpretation ----------------------- - :math:`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. - :math:`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. See also -------- .. seealso:: :doc:`abbreviations` Acronyms and short forms used across the project. :doc:`terms` Plain-language explanations of calibration, sharpness, identifiability, transfer, and other concepts.