Groundwater contour mapping in Venice by stochastic interpolators: 1. Theory

Abstract

An adaptation of the kriging technique (Matheron, 1969, 1970) is developed and used to map the hydraulic head field z of three major aquifers underlying the Venetian lagoon. Available records come from a fairly small number of observation wells unevenly scattered across the mainland, Venice, and the littoral. The event to be reconstructed is conceptually idealized as the sum of (1) a deterministic component m giving the main trend of z and (2) a stochastic component e giving the natural dispersion of z around m with zero mean, constant variance, and high autocorrelation. Both m and e are dependent on the observation scale. A physically based expression for m is selected including the consideration of additional information related to the general hydrogeological context. To reduce the complexity of the choice, some unknown coefficients a k are introduced in m and determined ‘a priori’ by a best fit technique, for instance, the least squares method. The relationship between m and a k is not necessarily linear. The dispersion e is then assessed by a linear optimal stochastic interpolator, i.e., by a linear combination of the observed values with the requirement that the variance of the interpolation error E be minimal. In the absence of instrument errors the interpolator reproduces the measured quantity in the observation points, while in the remainder of the field it provides an estimate z * with a standard deviation σ whose magnitude depends on the dispersion of the given z around m as well as on the quantity and distribution of the measurements. A prerequisite for a plausible model is that the reduced errors ( z − z * )/σ have zero mean and unit variance, i.e., the estimates are not affected by systematic errors and ( z − z * ) is consistent with the corresponding σ. Validation of the interpretation models selected for each of the three Venetian aquifers shows that the present approach yields accurate results provided the trend is correctly assessed.