Geochemical Modelling of Groundwater Using Multivariate Normal Distribution (MND) Theory

Abstract

Groundwater has become scarce especially in arid and semi-arid regions but is essential for drinking, domestic purposes and for survival of human beings. It must be conserved and not used for irrigation and industrial purposes especially in areas where no surface water is available. Groundwater is rarely polluted by natural processes but often by excess use of chemical fertilizers in agriculture and/or by disposal of untreated domestic/industrial wastes. Groundwater quality can be assessed using both physical (pH, EC, TDS etc.) and chemical characteristics (volume/weight concentrations of cations and anions measured on a sufficient water sample). Before use, groundwater must be tested for its required quality especially for drinking/domestic use. Quality of groundwater is assessed using linear statistical technology (multivariate normal distribution; MND) for measured continuous random variables on each sample. Prerequisites for applying MND theory are: (i) samples are independent and belong to one homogeneous population, and (ii) all input random variables possess Gaussian density. Since chemical constituents add to a constant sum (1.0 or 100%, 1000 (per mil), 1 million, 1 billion etc.) they are NOT INDEPENDENT as desired, but possess spurious negative correlations, as well as the random variables are not Gaussian but approximately log-normal with high positive skewness. Both these defects are simultaneously eliminated by log(c/(1 − c)) pre-transformation, where c (0 < c< 1) is the fractional concentration of any constituent. As chemical constituents (cations or anions) are found in trace quantities in groundwater, the log(c/(1 − c)) transform reduces to a simpler log (c) transform as inputs to MND model. The main MND methods are PCA and FA, Multiple Regression (Correlation) for a Single Populations and MANOVA, Linear Discriminant Functions (LDFs), MANCOVA for Multiple Populations. Geochemical models provide us with expected outcomes/correlations which can be compared/contrasted with observed outcomes/correlations to take appropriate and optimal decisions. Time series/geostatistical modeling requires very large number of samples along each line of investigation, hence these methods should not be used for routine groundwater studies.