First Section

- What is ordinary kriging? Let's study ordinary kriging definition, deviations of ordinary kriging equations, input data for ordinary kriging, steps of ordinary kriging, At the end, let's make a summary. Ordinary kriging is a variety of kriging which assumes that local means are not necessarily closely related to the population mean, and which therefore uses only the samples in the local neighborhood for the estimate. Equation one is a linear estimator. An estimator of the location Y* is the estimator of the value w alpha, for alpha equal to one to always applied to the data values of Y The bias is expectation of estimated value Y* minus the true value Y Plug in Equation Y for Y*, we get this part. Take the expectation for Yand Y, we get this part, it's expectation Yequal to mu, we get this part, take mu out, we get this part, so as long as sum of weight equal to one, the weighted linear estimator is unbiased. Equation two defines the error variance, which can be expanded as equation three, based on the square of a minus b equals to square of a minus 2ab plus square of b. Plug equation one into the first term and the second term in equation three. We get the first term and second term in equation four as expectation Y, Y equal to covariance , , we get the first term of the second term in equation five, as expectation Ysquared equal to covariance theory, that is variance and get the third term in equation five. Copy equation five here. The Lagrange multiplier is used for converting a constrained minimization problem into an unconstrained one by adding the third term into equation five. So, we get equation six. In order to minimize estimation variance, take the past derivatives of equation 6 with respect to each weight, we get equation seven. Set equation seven equal to zero. We get equation 8. There is a normal equation of ordinary kriging. Copy equation eight here and now copy the constrained weight equation here. We write an example of equation eight with the constraints for three data. It is shown in equation nine. Equation 10 and equation 11 are matrix formats for equation nine. Copy equation five and the equation eight here. A circular part in equation five and the circular part in equation eight are the same, we get equation 12. Equation 12 is ordinary kriging variance. They are shown in ordinary kriging result maps. Input data of conditional data, global mean, variogram parameters, search paramenters. There are 14 basic steps in the ordinary kriging process. One, is assign data into grid. Two, prepare variograms. Three, define search region. Four, run ordinary kriging for each undefined cell. Five, check results. In this example, our regional net-to-gross data, are x y locations and their values. Assign data into grid, which will be used as conditional data, seven conditioning data, are assigned to their nearest or greater centers. Three data are assigned into one cell. See blue circle. Multiple data are assigned to the same cell. We can take the average of them. The average of 0.8, 0.75, and 0.23 is 0.59. Update this value for this cell into 0.59, with the end one cell has one data. Prepare variogram model for this example, we get a spherical variogram. Semi major 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 30 degrees. It is the clockwise rotation angle of major axis from most spherical model with C zero equal to one. A zero will be defined wherever Y is at. Define search region into this eclipse. Select an un-estimated cell. Here, we select I equal to two, J equal to two. As ordinary kriging only uses the input data, the order of selecting undefined cell does not matter. Applying the search region is to put the search eclipse's center to the selected unestimated cell's center. Obtained data in search region's three data are obtained mark them as u one, u two, and u three. Also, we mark u. Let's review ordinary kriging equation. In order to solve weights, we need to know left side and right side are covariances. These covariances can be computed based on variogram model. For detail, refer to the lecture "How to apply variogram in kriging." After we get this ordinary kriging estimate and the ordinary kriging variance, can be obtained based on these two equations. Put the estimator 0.27 to the cell with I equal to two, J equal to 2. Input global mean is assigned to the cell when there are no conditional data obtained in the search area. Put the variances 0.03 to the cell with I equal to two, J equal to two. So variance is zero for the cell with contour plotted data. See five zeros for five contour plotted data. Zero is assigned to the cell when there are no conditional data obtained in the search area. The similar procedures as the first cell will be conducted on all cells. Now we have finished all the steps. We need to check if results are reasonable. If results honor input conditional data. If results honor the input variogram model. Let's make a summary. The ordinary kriging estimation is unbiased as the sum of beta is equal to one. The ordinary kriging estimation honors the actually observed values. It generates ordinary kriging estimate and ordinary kriging variance, which provides 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 generates deterministic results. Ordinary kriging is the most commonly used method for environmental situations. This is the end of the lecture.