Eulogio Pardo-Igúzquiza

Optimal areal rainfall estimation using raingauges and satellite data

Abstract The main aim of this paper is to present a new method of areal rainfall estimation using satellite and ground-based data. This method involves optimal merging of the estimates provided by satellite information and estimates obtained from raingauges. In the merging procedure, each estimate is weighted according to its uncertainty given by its estimation variance. The uncertainty attributed to the raingauge estimates is obtained using block kriging, while for the satellite uncertainties, a novel regression approach is developed. A standard error is also attached to the new merged estimates. In order to test the algorithm, a case study has been undertaken using the EPSAT dense raingauge network in Niger. The complete EPSAT raingauge network (94 gauges distributed over a 1×1° square) has been used to obtain a detailed picture of the rainfall pattern which is then used as a reference for comparing the estimation schemes. The schemes compared are: (1) estimates based on satellite data only; (2) kriged estimates from a randomly selected subset of four gauges; (3) kriging with external drift using both satellite data and the subset of gauges; and (4) the new merging algorithm. The merging process gives more reliable results both for the mean areal rainfall and its spatial distribution.

Optimal selection of number and location of rainfall gauges for areal rainfall estimation using geostatistics and simulated annealing

This paper presents a method for establishing an optimal network design for the estimation of areal averages of rainfall events. The problem consists of minimising an objective function which includes both the accuracy of the areal mean estimation (as expressed by the kriging variance of estimation) and the economic cost of the data collection. The well known geostatistical variance-reduction method is used in combination with simulated annealing as an algorithm of minimisation. This methodology has several advantages which will be demonstrated in this paper. Several synthetic examples are shown in order to illustrate the performance of the methodology in two different optimisation problems: the optimal selection of a subset from a set of stations that already exist and the optimal augmentation of a previously existing network.

The Fourier Integral Method: An efficient spectral method for simulation of random fields

The Fourier Integral Method (FIM) of spectral simulation, adapted to generate realizations of a random function in one, two, or three dimensions, is shown to be an efficient technique of non-conditional geostatistical simulation. The main contribution is the use of the fast Fourier transform for both numerical calculus of the density spectral function and as generator of random finite multidimensional sequences with imposed covariance. Results obtained with the FIM are compared with those obtained by other classic methods: Shinozuka and Jan Method in 1D and Turning Bands Method in 2D and 3D, the points for and against different methodologies are discussed. Moreover, with the FIM the simulation of nested structures, one of which can be a nugget effect and the simulation of both zonal and geometric anisotropy is straightforward. All steps taken to implement the FIM methodology are discussed.

Comparison of geostatistical methods for estimating the areal average climatological rainfall mean using data on precipitation and topography

The results of estimating the areal average climatological rainfall mean in the Guadalhorce river basin in southern Spain are presented in this paper. The classical Thiessen method and three different geostatistical approaches (ordinary kriging, cokriging and kriging with an external drift) have been used as estimators and their results are compared and discussed. The first two methods use only rainfall information, while cokriging and kriging with an external drift use both precipitation data and orographic information (easily accessible from topographic maps). In the case study presented, kriging with an external drift seems to give the most coherent results in accordance with cross-validation statistics. If there is a correlation between climatological rainfall mean and altitude, it seems logical that the inclusion of topographic information should improve the estimates. Kriging with an external drift has the advantage of requiring a less demanding variogram analysis than cokriging.

Downscaling Cokriging for Super-Resolution Mapping of Continua in Remotely Sensed Images

The main aim of this paper is to show the implementation and application of downscaling cokriging for super-resolution image mapping. By super-resolution, we mean increasing the spatial resolution of satellite sensor images where the pixel size to be predicted is smaller than the pixel size of the empirical image with the finest spatial resolution. It is assumed that coregistered images with different spatial and spectral resolutions of the same scene are available. The main advantages of cokriging are that it takes into account the correlation and cross correlation of images, it accounts for the different supports (i.e., pixel sizes), it can explicitly take into account the point spread function of the sensor, and it has the property of prediction coherence. In addition, ancillary images (topographic maps, thematic maps, etc.) as well as sparse experimental data could be included in the process. The main problem is that super-resolution cokriging requires several covariances and cross covariances, some of which are not empirically accessible (i.e., from the pixel values of the images). In the adopted solution, the fundamental concept is that of covariances and cross-covariance models with point support. Once the set of point-support models is estimated using linear systems theory, any pixel-support covariance and cross covariance can be easily obtained by regularization. We show the performance of the method using Landsat Enhanced Thematic Mapper Plus images.

Maximum Likelihood Estimation of Spatial Covariance Parameters

In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters.

MLREML: a computer program for the inference of spatial covariance parameters by maximum likelihood and restricted maximum liklihood

Abstract Maximum likelihood and restricted maximum likelihood are appealing parametric alternatives to other classical nonparametric methods to estimate the covariance parameters of spatial variables. MLREML, an ANSI FORTRAN-77 computer program which performs both kinds of inference methods is presented. One important characteristic of these methods is that they provide a measure of the uncertainty of the estimates. Two datasets are provided to check the implementation of the program and the results are discussed.

Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.

CONNEC3D: a computer program for connectivity analysis of 3D random set models

Geostatistical simulation is used in risk analysis studies to incorporate the spatial uncertainty of experimental variables that are significantly under-sampled. For example, the values of hydraulic conductivity or porosity are critical in petroleum reservoir production modelling and prediction, in assessing underground sites as waste repositories, and in modelling the transport of contaminants in aquifers. In all these examples connectivity of the permeable phase or permeable lithofacies is a critical issue. Given an indicator map on a regular two- or three-dimensional grid, which can be obtained from continuous-valued or from categorical variables, CONNEC3D performs a connectivity analysis of the phase of interest (coded 0 or 1 by an indicator function). 3D maps of multiple indicators, categories or continuous variables can also be analysed for connectivity by suitable coding of the input map. Connectivity analysis involves the estimation of the connectivity function τ(h) for different spatial directions and a number of connectivity statistics. Included in the latter are the number of connected components (ncc), average size of a connected component (cc), mean length of a cc in the X, Y and Z directions, size of the largest cc, maximum length of a cc along X, Y and Z and the numbers of percolating components along X, Y and Z. In addition, the program provides as output a file in which each cc is identified by an integer number ranging from 1 to ncc. The implementation of the program is demonstrated on a random set model generated by the sequential indicator algorithm. This provides a means of estimating the computational time required for different grid sizes and is also used to demonstrate computationally that when the semi-variogram of the indicator function is anisotropic the connectivity function is also anisotropic. There are options within the program for 6-connectivity analysis, 18-connectivity analysis and 26-connectivity analysis. The software is provided in two formats, as a stand-alone program that can perform connectivity analysis of an indicator map and as a subroutine that can be repeatedly called in order to calculate averages of connectivity analyses of a large number of realizations of indicator maps, or to identify critical realizations generated by conditional simulations of continuous variables or categorical variables.

Downscaling Landsat 7 ETM+ thermal imagery using land surface temperature and NDVI images

Thermal infrared (TIR) satellite images and derived land surface temperature (LST) are variables of great interest in many remote sensing applications. However, the TIR band has a spatial resolution which is coarser than the other multispectral bands for a given satellite sensor (visible, near and shortwave infrared bands); therefore, the spatial resolution of the retrieved LST from available satellite-borne sensors is not accurate enough to be used in certain applications. The application of a method is shown here for obtaining LST images with enhanced spatial resolution using the LST at a coarser resolution and the Normalized Difference Vegetation Index (NDVI) of the same scene using Downscaling Cokriging (DCK). A LST image with perfect coherence was obtained by applying this method to a Landsat 7 ETM+ image. This implies that, if the downscaled LST image is degraded to its original resolution, the degraded image obtained is identical to the original. Hence high spatial resolution LST images were obtained without altering the original radiometry with the inclusion of artefacts. Moreover, the performance of DCK was compared with global and local TSHARP methods. The RMSE of the sharpened images were 0.85, 0.92 and 1.1 K, respectively.