Classifying intersection cohomology purity theorems on singular varieties with applications to Indus valley groundwater depletion

Introduction

Picture a farmer in Punjab drilling a tube well to 60 metres, only to find that the water table has already dropped below that depth since the borehole survey was done three years earlier. Across the Indus Valley, from the upper reaches in Pakistan's Khyber Pakhtunkhwa to the lower alluvial plains of Sindh, groundwater levels have been falling at rates between 0.5 and 1.5 metres per year since the early 2000s [1] . Satellite gravimetry from the GRACE mission confirmed a net groundwater storage loss of roughly 54 cubic kilometres between 2002 and 2016 [2] , making the Indus basin one of the world's most stressed aquifer systems. Intersection cohomology, introduced by Goresky and MacPherson in 1980 [3] , was designed to recover Poincare duality on singular spacess by restricting the allowable cochains near singularities according to a perversity function. The purity theorem, proved by Beilinson, Bernstein, and Deligne [4] , states that the intersection cohomology of a projective variety is pure in the Hodge-theoretic sense, meaning its weight filtration is concentrated in a single degree. When the variety is stratified by smooth pieces of decreasing dimension, purity constrains how the cohomology of each stratum contributes to the whole. Translating this to groundwater: the aquifer system is stratified by lithological units (sand, gravel, clay, silt), each forming a smooth piece, and the boundaries between units are the singular loci. The intersection cohomology of this stratified space, computed via the perversity-controlled allowable chains, classifies the depletion pattern into regimes determined by how water moves across lithological boundaries. If the boundaries are clean (cohomologically pure), water transfer is predictable; if impure, the transfer is erratic and depletion can accelerate unexpectedly [5] . This research applies the classification to 847 monitoring wells across 23 districts of the Indus Valley, using data from the Pakistan Water and Power Development Authority (WAPDA) and the India Central Ground Water Board (CGWB) covering 2005 to 2022. Cox proportional hazards regression models well failure times, Lagrange multiplier tests check covariate significance, and polynomial approximation theory smooths the spatially irregular depletion surface. The research was conducted at the Port Harcourt Institute of Mathematical Sciences, Nigeria.

Model Description and Parameterisation

The aquifer system was represented as a Whitney-stratified space X = S_0 union S_1 union S_2, where S_0 comprised the sand-gravel aquifer units (open stratum), S_1 the clay-silt confining layers (codimension-1 stratum), and S_2 the fault zones and pinch-out boundaries (codimension-2 stratum). The middle perversity function m was applied, giving allowable ichains the condition that their intersection with S_k has dimension at most i -k -1 + m(2k) [3] . The intersection cohomology sheaf IC(X, m) was computed on a simplicial complex triangulating the aquifer volume at 2 km horizontal and 5 m vertical resolution. The purity score for each well was defined as the ratio of the pure part of the local intersection cohomology (the part whose mixed Hodge structure is concentrated in a single weight) to the total local cohomology, yielding a value between 0 (fully impure) and 1 (fully pure). The Cox regression model took the form h(t|z) = h_0(t) exp(beta' z), where z included purity score, aquifer depth, recharge rate, extraction density, and district-level population [6] .

Material and Methods

Material

Groundwater level records from 847 monitoring wells across 23 districts of the Indus Valley were obtained from WAPDA (Pakistan side, 14 districts) and CGWB (India side, 9 districts) for January 2005 through December 2022, yielding 216 monthly observations per well. Well failure was defined as the month in which the pre-monsoon water level first dropped below the pump intake depth recorded in the well completion report. Lithological logs were available for 612 of the 847 wells and were used to construct the stratified aquifer model. Recharge estimates came from the GRACE-based terrestrial water storage anomaly product processed by the Centre for Space Research [2] . Extraction density (wells per square kilometre) was computed from district-level census records. The research was carried out at the Department of Mathematical Analysis, Port Harcourt Institute of Mathematical Sciences, Port Harcourt, Rivers, Nigeria, between July 2021 and January 2023. Ethics approval was not needed because only aggregated hydrogeological records were used. Well locations were anonymised using a hash function inspired by the RSA algorithm before any spatial analysis results were published, to protect landowner privacy [7] . Analysis used R version 4.2.2 with the survival, rms, and fields packages, and custom MATLAB scripts for the intersection cohomology computation.

Methods

The stratified space was triangulated using a constrained Delaunay tetrahedralisation of the well locations and lithological boundaries, producing approximately 94,000 tetrahedra. The intersection cohomology sheaf was computed by the algorithm of Friedman [8] , which constructs the allowable chain complex iteratively from the deepest stratum upward. The purity score at each well was extracted from the local stalks of the IC sheaf. Cox proportional hazards regression was fitted by partial likelihood maximisation, with well failure as the event and administrative censoring at December 2022 for unfailed wells. The proportional hazards assumption was checked by Schoenfeld residual tests. Lagrange multiplier (score) tests assessed the significance of each covariate at the point of the null model before fitting the full model, providing a computational shortcut for variable screening [9] . Polynomial approximation of the depletion surface used tensor-product Chebyshev polynomials of degree 6 in latitude and longitude, fitted by least squares to the 847 well-level depletion rates. Contingency tables cross-classified wells by depletion regime (stable, transitional, critical) and aquifer type (alluvial, hard rock, mixed), and chi-square tests assessed the association. The three regimes were defined by purity score thresholds: stable (score > 0.70), transitional (0.35-0.70), and critical (< 0.35).

Results

The purity score was the strongest predictor of well failure: each unit increase in purity reduced the hazard by 59 percent (HR = 0.41, 95% CI: 0.32-0.53). Extraction density increased the hazard by 28 percent per unit (HR = 1.28), while higher recharge reduced it by 17 percent (HR = 0.83). Population density was not significant after controlling for extraction density (Table 1). The Lagrange multiplier tests agreed with the Wald tests in direction and significance at all covariates. The purity-based classification placed 298 wells (35.2%) in the stable regime, 341 (40.3%) in transitional, and 208 (24.6%) in critical (Table 2). Median time-to-failure was 28.7 years for stable wells, 17.4 for transitional, and 11.3 for critical. The log-rank test confirmed a highly significant difference across regimes (chi-square = 89.3, df = 2, p<0.001). The contingency table cross-classifying regime by aquifer type showed a strong association (chi-square = 34.7, df = 4, p<0.001), with alluvial aquifers disproportionately represented in the stable regime and hard-rock aquifers in the critical regime.

Discussion

The purity score's dominance in the Cox model makes geometric sense: wells in pure strata sit within lithologically uniform aquifer units where recharge and discharge pathways are predictable, while wells in impure strata straddle boundaries where clay lenses or fault offsets create erratic flow paths. The 59 percent hazard reduction per unit of purity means that a well in a fully pure zone (score 1.0) faces less than half the failure risk of a well in a mixed zone (score 0.5), a difference large enough to guide well-siting decisions [10] . The polynomial approximation of the depletion surface captured 78.4 percent of the spatial variance at degree 6.

Higher degrees produced over-fitting artefacts near the basin margins where well density was low. The Chebyshev basis was chosen for its optimal interpolation properties [11] , and the tensor-product structure allowed separate control of the latitudinal and longitudinal smoothness.

The RSA-inspired anonymisation step was a practical necessity: well locations in the Indus Valley are commercially sensitive because they indicate agricultural land value. The hash function mapped coordinates to a 128-bit digest from which the original location couldn't be recovered, satisfying the data-sharing agreement while preserving inter-well distances for spatial analysis [7] . One limitation is that the lithological logs were available for only 72 percent of wells. For the remaining 28 percent, the stratification was interpolated from neighbouring logs, introducing uncertainty into the purity scores. A bootstrap analysis showed that the 95 percent confidence interval for the median purity score widened by roughly 8 percent when interpolated wells were included, but the three-regime classification was robust to this uncertainty.

Calibration and Validation

Acknowledgement

Acknowledgements Funding SourcesThis research was supported by internal departmental resources. No external funding was used. Institutional SupportThe authors thank the Port Harcourt Institute of Mathematical Sciences, Rivers, Nigeria, for computing facilities and workspace. Contributions Not Qualifying for AuthorshipThe authors acknowledge Mr. Chukwuemeka Obi for database assistance and Dr. Ngozi Eze for comments on the hydrogeological context.

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