Estimating the variance and integral scale of the transmissivity field using head residual increments

A modification of previously published solutions regarding the spatial variation of hydraulic heads is discussed whereby the semivariogram of increments of head residuals (termed head residual increments HRIs) are related to the variance and integral scale of the transmissivity field. A first‐order solution is developed for the case of a transmissivity field which is isotropic and whose second‐order behavior can be characterized by an exponential covariance structure. The estimates of the variance σ_Y² and the integral scale λ of the log transmissivity field are then obtained via fitting a theoretical semivariogram for the HRI to its sample semivariogram. This approach is applied to head data sampled from a series of two‐dimensional, simulated aquifers with isotropic, exponential covariance structures and varying degrees of heterogeneity (σ_Y² = 0.25, 0.5, 1.0, 2.0, and 5.0). The results show that this method provided reliable estimates for both λ and σ_Y² in aquifers with the value of σ_Y² up to 2.0, but the errors in those estimates were higher for σ_Y² equal to 5.0. It is also demonstrated through numerical experiments and theoretical arguments that the head residual increments will provide a sample semivariogram with a lower variance than will the use of the head residuals without calculation of increments.