- What is point estimation? Let's study polygonal estimator, triangulation estimator, inverse distance estimator, also search consideration. At the end, let's make a summary. Point estimation is to estimate unknown values at specific locations using weighted linear combinations of known data. The weights need to account not only for possible clustering but also for the distance to the nearby samples. Input data are conditional data, usually well log data. Here is an example of seven data points shown by x-y coordinates and the value. We will estimate the values at given locations. Here is an example. To estimate this value at x equal to 65, y equal to 137, see red question mark. Polygon method gives all the weight to the closest sample value, that is the nearest neighbor value. Find out the sample of the closest neighbor, d is distance. Here, the closest distance is 3.6. Given the value of the closest location here is 696, and assign it to the to-be-estimated location. Let's take a look at this diagram. As long as the points we are estimating fall within the same polygon of the influence, the polygonal estimate remains unchanged. As soon as we encounter a point in a different polygon of influence, the estimate jumps to a different value. So the final results is discontinuous surface of plateaus. Triangulation method is to estimate unknown using its three nearby points, which consist of a triangle. How to find out these points for triangles? For a set of points in 2-D, Delaunay triangulation of these points ensures the circumcircle associated with each triangle contains no other point in its interior. This is the example of Delaunay triangulation for our seven data points. Let's go back to estimate a value at x is equal to 65, y is equal to 137. First copy three vertex data into the map. Make triangle OJK. Make triangle OIK. Make triangle OIJ. Compare the area of triangle for each of them. They are 9.5, 12, and 22.5, shown in this figure. The estimated value at location O is a weighted linear combination of values at three vertexes, in which each value is weighted according to the area of the opposite triangle. VI is value at I, AOJK is the area of triangle of OJK. VJ is value at J, AOIK is the area of triangle of OIK. VK is value at K, AOIJ is the area of triangle of OIJ. Here is an example applying the above equation. The triangle OJK area 22.5. The value at the vertexes I is 696. It is similar for another two triangles. The estimated result is 570.19. Inverse distance method gives more weight to the closest samples and less to those that are farthest away. It makes the weight for each sample inversely proportional to its distance from the point being estimated. It's shown in the equation, where d1 to dn are the distances from each of the n samples in the point being estimated, and v1 to vn are the sample values. In more general estimation, d can be powered to p, usually it is from one to three. Extremely cases, when p equal to zero, this estimation is simple average method. When p equal to infinity, this estimation is polygon method. Here is an example for inverse distance method. The estimated result is 594. Search region can be applied to control which samples are included in the estimation. Let's make a summary. Simplistic method to do point estimation is arithmetic average using equal weights for all samples. This lecture addresses three methods to do point estimation. They are polygon method, triangulation method, and inverse distance method. For all method, search region can be applied. For details, refer to book titled, An Introduction to Applied Geostatistics. That's the end of the lecture.