First Section

- Let's study. Simple Kriging definition. Deviations of simple Kriging equations. Input data for simple Kriging. Steps of simple Kriging. At the end, let's make a summary. Simple Kriging is defined as a variety of Kriging that assumes that local means are relatively constant and equal to the population mean, which is well known. The population mean is used as a factor in each local estimate along with the samples in the local neighborhood. Residual data values, Y u alpha are defined as Z u alpha minus m. Z u alpha for alpha equal to one to n are n data values, m is global mean. Equation one is a linear estimator which has estimated location, Y star u, is estimated value, W alpha for alpha equal to one to n always applied to the n residual data values of Y u alpha. Equation two defines the error variance which can be extended in equation three based on square of a minus b equal to square of a minus two a b plus square of b. Plug equation one into the first term and the second term in equation three. We get the first term and the second term in equation four as expectation Y u beta, Y u alpha equal to covariance u beta, u alpha, we get the first term and the second term in equation five as expectation Y u square equal to covariance zero. That is the variance. Get a sub term in equation five. In order to minimize estimation variance, take the partial derivatives of equation five with respect to each weights, w. We get equation six. Set equation six equal to zero. We get equation seven. They are the normal equations of simple Kriging. Copy equation seven here. Look at example for three data in equation eight, equation nine and equation 10 are matrix format for equation eight. Copy equation five and equation seven here as circled part in equation five and circled part in equation seven are the same. We get equation 11. Equation 11 is simple Kriging variance. They are shown in simple Kriging result maps. Input data are conditional data, global mean, variogram parameters, and search parameters. There are following basic steps in the simple Kriging process. One, assign data into grid. Two, prepare variograms. Three, define search region. Four, run simple Kriging for each undefined cell. Five, check results. In this example, original net-to-gross data are x y locations under values. Average of data is 0.4. Residual is data minus average. Assign data into grid which we will use as conditioning data. Seven conditioning data are assigned to their nearest greatest cell centers. Three data are assigned into one cell. See blue circles. Where multiple data are assigned to the same cell, we can take the average of them. The average of 0.4, 0.35 and minus 0.17 is 0.19. Update the value for this cell into 0.19 for the n. One cell has one data. Prepare variogram model for this example. We get anvariogram, semi variogram axis equal to 25. It is the correlation range with the maximum continuity. Semi minor axis equal to 15. It is the correlation range perpendicular to the major axis with minimum continuity. Azimuth equal to thirty degrees. It is the clock-wise rotation angle of major axis Spherical model with C zero equal to one. A zero will be defined wherever use it. Define search region in 2D. It is an eclipse. Select an un-estimated cell. Here, we select i equal to two, j equal to two. A simple Kriging, only uses the input data. So order of selecting un-estimated cell does not matter. Apply your search region is to put the search eclipse center to the selected un-estimated cell center. Obtain data in search region. Three data are obtained. Mark them as u one, u two and u three also. Remark, let's review simple Kriging equation. In order to solve weights, we need to know left side and the right side convergences. This convergences can be computed based on variogram model For details,is a lecture. How to apply variogram in Kriging. After we get weights, simple Kriging estimate, and the simple Kriging variance can be obtained based on these two equations. Put the Kriging estimator, 0.17 to the cell, i equal to two, j equal to two. Zero is a sign, it was a cell where there are no conditional data obtained in the search area. For the Kriging variance, 0.03 to the cell, with i equal to two, g equal to two. So variance is zero for the cell. With control upon the data. See, five zeroes for five control data pods. Zero is assigned to the cell where there are no conditional data assigned in the search area. The similar procedures as the first cell will be conducted on all cells at global mean. Back to the final result. Now we have finished all cells. We need to check. If results are reasonable. If results honor input conditional data. If results honor the input global mean. If results honor the input variogram model. Let's make a summary. The simple Kriging estimation is unbiased. The simple Kriging estimation honors the actually observed values. It generates simple Kriging estimate and simple Kriging variance. They should provide a measure of precision. It only uses input data when computing a value at an un-estimated grid cell. It is a non-parametric method of interpolation. It generated deterministic results. This is not usually the most appropriate method for environmental situations. This is the end of the lecture.