The point-estimation methods described in this section consist of common methods used to make contour maps. These methods use non-geostatistical interpolation algorithms and do not require a spatial model. They provide a way to create an initial “quick look” map of the attributes of interest. This section is not meant to provide an exhaustive dissertation of the subject, but will introduce certain concepts needed to understand the principles of geostatistical interpolation and simulation methods discussed in later sections.
Most interpolation methods use a weighted average of values from control points in the vicinity of the grid node in order to estimate the value of the attribute assigned to that node. With this approach, the attribute values of the nearest control points are weighted according to their distance from the grid node, with the heavier weights assigned to the closest points. The attribute values of grid nodes that lie beyond the outermost control points must be extrapolated from values assigned to the nearest control points.
Many of the following methods require the definition of Neighborhood parameters to characterize the set of sample points used during the estimation process, given the location of the grid node. For the upcoming examples, we‟ve specified the following neighborhood parameters:
Isotropic ellipse with a radius = 5000 feet
4 quadrants
A minimum of 7 sample points, with an optimum of 3 sample points per quadrant
These examples use porosity measurements, located on a nearly regular grid.
See Figure 1 (Location and values of control points within the mapping area at North Cowden Field, West Texas) for the sample locations and porosity values.
Figure 1
The following seven estimation methods will be discussed in turn:
Inverse Distance
Closest Point
Moving Average
Least Squares Polynomial
Spline
Polygons of Influence
Triangulation
The first five estimation methods are accompanied by images that illustrate the patterns and relative magnitude of the porosity values created by each method.
All images have the same color scale. The lowest value of porosity is dark blue (5%) and the highest value is red (13%), with a 0.5% color interval. However, for the purpose of this illustration, the actual values are not important at this time.
(No porosity mapping images were produced for the polygons of influence and triangulation methods.)
INVERSE DISTANCE
This estimation method uses a linear combination of attribute values from
neighboring control points. The weights assigned to the measured values used in the interpolation process are based on distance from the grid node, and are inversely proportional, at a given power (p). If the smallest distance is smaller than a given threshold, the value of the corresponding sample is copied to the grid node. Large values of p ( 5 or greater) create maps similar to the closest point method (Isaaks and Srivastava, 1989; Jones, et al., 1986), which will be described next. The equation for the inverse distance method has the following form:
Figure 2 displays Inverse Distance gridding using a power of 1.
Figure 2
The Inverse Distance method is recommended as a “first pass” through the data because it:
is simple to use and understand.
produces a “quick map.”
is an excellent QC tool.
locates “bulls-eye” effect (lone high or low values).
spots erroneous sample locations.
gives a first indication of trends.
CLOSEST POINT
The closest point (Figure 3) or nearest neighbor methods consist of copying the value of the closest sample point to the target grid node.
Figure 3
This method can be viewed as a linear combination of the neighboring points with all the weights equal to 0, except the weight attached to the closest point which is equal to 100% (Henley, 1981; Jones, et al., 1986).
Z* = Z (closest
Where
Z* = the target grid node location Z = the data points
MOVING AVERAGE
The moving average method (Figure 4) is the most frequently used estimation method.
Figure 4
Each neighboring sample point is given the same weight. The weight is calculated so that the sum of the weights of all the neighboring sample points sum to unity (Henley, 1981; Jones, et al., 1986).
So, if we assume that there are N neighboring data, Z* = Z /N
Where
Z* = the target grid node location Z = the data points
N = the neighboring data
The moving average takes its name from the process of estimating the attribute value at each grid node based on the weighted average of nearby control points in the search neighborhood, and then moving the neighborhood from grid node to node.
LEAST SQUARES POLYNOMIAL
The least squares polynomial method (Figure 5) is commonly used for trend surface analysis.
Figure 5
The neighboring points are used to fit a polynomial expression of a degree specified by the user. The polynomial form is a logical choice for surface approximation, as any function that is continuous and possesses all derivatives can be reproduced by an infinite power series. The polynomial surface is a mathematical function involving powers of X and Y. The complexity of the surface (Table 3.1) is controlled by the user through the number of terms used, which is dependent upon its degree, N, a positive integer (Jones, et al., 1986; Davis, 1986; Krumbein and Graybill, 1965, Henley, 1981).
Z* = aij Xi Yj Where
Z* = the target grid node location
Table 1: General form of polynomial functions (after (Jones, ET al., 1986).
Degree
Spline fitting is a commonly used quantitative method. The method ignores geologic trends and allows sample location geometry to dictate the range of influence of the samples. The bicubic spline (Figure 6) is a two-dimensional gridding algorithm.
Figure 6
In one-dimension, the function has the form of a flexible rod between the sample points. In two dimensions, the function has the form of a flexible sheet. The objective of the method is to fit the smoothest possible surface through all the samples using a least squares polynomial approach (Jones, et al., 1986).
POLYGONS OF INFLUENCE
This is a very simple method, and is often used in the mining industry to estimate average ore grade within blocks. Often, the value estimated at any location is simply the value of the closest point. The method is similar to the closest point approach. Polygonal patterns are created, based on sample location. Polygon boundaries represent the distance midway between adjacent sample locations.
As long as the points we are estimating fall within the same polygon of influence, the polygonal estimate does not change. As soon as we encounter a grid node in a different polygon, the estimate changes to a different value. This method causes abrupt discontinuities in the surface, and may create unrealistic maps (Isaaks and Srivastava, 1989; Henley, 1981).
TRIANGULATION
The triangulation method is used to calculate the value of a variable (such as depth, or porosity for instance) in an area of a map located between 3 known control points. Triangulation overcomes the problem of the polygonal method, removing possible discontinuities between adjacent points by fitting a plane through three sample points that surround the grid node being estimated (Isaaks and Srivastava, 1989). The equation of the plane is generally expressed as:
Z* = ax + by + c
This method starts by connecting lines between the 3 known control points to form a triangle (denoted as rst in Figure 7).
Figure 7
Next, join a line from the unknown point (point O in the figure), to each of the corners of the triangle, thereby forming 3 new triangles within the original triangle.
The value of any point located within the triangle (point O in this example) can be determined through the following steps:
1. Compute the areas of the resulting new triangles
2. Use the areas of each triangle to establish a weight for each corner point 3. Multiply the values of the three corner points by their respective weights,
and
4. Add the resulting products.
The formula to find the area of a triangle is:
A = bh/2 Where
A is the area of the triangle, b is the length of the base, and
h is the length of the height, taken perpendicular to the base.
Weights are assigned to each value in proportion to the area of the triangle opposite the known value, as shown by the example in the Figure 7. This example shows how the values from the three closest locations are weighted by triangular areas to form an estimated value at point O. The control values (r,s,t) are located at the corners of the triangle.
The value at point r is weighted by the triangular area AOst,
point s is weighted by the area AOrt, and
point t is weighted by the area AOrs.
The weights are taken as a percentage, where the sum of all 3 weights equals 1.
Now multiply the weight times its associated control value to arrive at a weighted control value. Do this for each of the three points. Then add up the 3 weighted control values to triangulate an interpolated depth for point O.
Be aware, however, that choosing different meshes of triangles or entering the data in a different sequence may result in a different set of contours for your map.
MAP DISPLAY TYPES