A Modified Discrete Karhunen-Loeve Random Field Estimator
Based upon the modified Karhunen-Loeve model of the gravity disturbance vector proposed by us recently, a corresponding Karhunen-Loeve random field estimator is developed in this paper for a two-dimensional grid of gravity data in a local finite region.The new eigenvectors obey the "required" orthogonality relations on the chosen grid provided the KL expansion is now separated into odd and even integers. The closed-form solution of the estimator isthen obtained under the "diagonal" assumption for the gain coefficients. It is shown that this assumption, without which nosolution is possible, enables the estimator to reproduce the data atgrid points when the noise in the data is absent and the number of terms in the Karhunen-Loeve expansion are equal to the number of grid points.
Error Covariance Analysis of a Karhunen-Loeve Random Field Estimator
An error covariance analysis of a two-dimensional Karhunen-Loeve random field estimator (KLE) for gridded gravity data is presented without actually using the data. Attention is focused particularly on the north-south deflection component of the disturbance vector so that the rms value of its residual error may not exceed 3 percent of the signal rms for grid spacing of not less than 5 nm, when signal-to-noise ratio in the data is varied within reasonable limits. To achieve this rather stringent goal, a discrete KLE derived entirely from a two-dimensional grid configuration is needed. The KLE for geodesy applications was developed initially by Bose. Using continuous measurements and the orthogonality relations of the Karhunen-Loeve (KL) eigenvectors in a continuous domain, continuous KL gain coefficients ßmnj and consequently the continuous KL gain vector Kmn were obtained; thus a satisfactory continuous KLE was achieved. However the discrete version of the KLE was then given by keeping the old (continuous) ßmnj, and simply computing the KL eigenvectors on the grid coordinates to evaluate the discrete KL gain vector Kmn. Clearly this is not a satisfactory discrete KLE since , ßmnj are not constants but rather depend on the measurement noise variances. So if the measurements are now going to be discrete, one must rederive ßmnj for this situation.
Experimental study of water fluxes in a residential area: 2. Road infiltration, runoff and evaporation
Lack of accurate data has led some hydrologists and city planners to assume that urban infiltration is zero and runoff is 100% of the rainfall. These assumptions lead to an over estimation of road runoff volume and an underestimation of direct recharge to groundwater, which is already rising under some UK cities. This study investigates infiltration and runoff processes and quantifies the percentage of rainfall that contributes to storm drainage, and that which infiltrates through different types of road surface. Access tubes were installed for measuring soil water content using a neutron probe in three car parks, a road and a grass site at the Centre for Ecology and Hydrology, Crowmarsh Gifford, Wallingford. Storm drainage was recorded at the exit of the Thamesmead Estate in Crowmarsh Gifford, just before the drain joins the River Thames at Wallingford. Rainfall and water table depth were also recorded. Weekly measurements of soil moisture content indicated that the top 40 cm layer is not influenced by water‐table fluctuations and, therefore, positive changes in soil moisture could be attributed to infiltration of rainfall through the surface. Depending on the nature of the surface, subsurface layers, level of traffic, etc., between 6 and 9% of rainfall was found to infiltrate through the road surfaces studied. The storm drainage generated by road runoff revealed a flow pattern similar to that of the receiving watercourse (River Thames) and increased with the increase of infiltration and soil water content below the road surface. The ratio of runoff to rainfall was 0·7, 0·9 and 0·5 for annual, winter (October–March) and summer (April–September) respectively. As the results of the infiltration indicated that 6 to 9% of annual rainfall infiltrates through the road surface, this means that evaporation represents, 21–24% of annual rainfall, with more evaporation taking place during summer than winter. Copyright © 2003 John Wiley & Sons, Ltd.