4.
Estimation
Harnanti Yogaputri Hutami, M.T
Geostatistika 2020
The Objective
• We are no longer interested in describing the sample
data set.
• Predicting values in the unsampled areas using the
sample informations available.
Features of an estimation
problem
• Do we want a global or local estimate?
• Do we want to estimate only the mean or the
complete distribution of data values?
• Do we want estimate of point values or of larger block
values?
Local Estimation Mean
• The Goal of Estimation is to Obtain the Single “Best” Value at an Unsampled
Location
• In Practice, the Estimated Value is a Function of the Algorithm (Model) Used
• Many Algorithms Have Been Developed; Each Have Advantages and
Disadvantages Compared to the Others
• All Estimation Methods Involve a Weighted Linear Combination of Sample Data
Values. That is,
z* = li z( xi )
n
i =1
where z(xi) = Sample Data Value at Location zi, li = Weight Assigned to z(xi),
and z* = Estimated Value at Location zi
Local Estimation Mean
• Estimation Algorithms Can Be Classified In Many Ways. One Useful Classification
Is
• Global Estimation
• Estimate Value Over Large Area (Volume)
• Local Estimates
• Global Estimates
• Consider Data Within Area (Volume) to Be Estimated
• Point Estimation
• Estimate Value Over Small Area (Volume)
• Point Values
• Block Values
• Consider Data Outside Area (Volume) to be Estimated
Local Estimation Mean
• Point Estimation Methods
• Geological Experience and/or Artistic License
• Traditional Algorithms That Use Weights Based on Euclidean (Geometric) Distance
• Polygon Method (Nearest Neighbor)
• Triangulation
• Local Sample Mean
• Inverse Distance
• Geostatistical Algorithms That Use Weights Based on “Structural” (or Statistical) Distance
• Simple Kriging
• Ordinary Kriging
• Universal Kriging
• Kriging with Trend
• Collocated Cokriging
Local Estimation Mean
• Problems Affecting All Point Estimation Methods
• How to Weight Samples
• Search Neighborhood
• Data Clustering
• For All Point Estimation Methods:
• Estimate (z*) is a Weighted Linear Combination
l z( x )
n
z* = i i
i =1
• Unbiasedness Condition Generally Given by
n
l i =1
i=1
Example Data Set
Well X Y Value Distance to P 140 +2 +5 +6
1 61 139 477 4,47 +1 696 606 791
477
2 63 140 696 3,61 *P
???
3 64 129 227 8,06
4 68 128 646 9,49
5 71 140 606 6,71 130
+3 +4
6 73 141 791 8,94 227 646
+7
783
7 75 128 783 13,45
60 70 80
1. Polygon (Nearest
Neighbor) Method 140 +2 +5
+6
+1 791
696 606
Polygon (Nearest Neighbor) Method 477
*X
• Assign All Weight To Nearest ???
Neighbor (Well 2 In This Case)
• Use Perpendicular Bisectors to
130
Divide Into Regions. Note that +3 +4 +7
227 646 783
X Is Closest to Well 2
➔ Estimated Value = 696
60 70 80
1. Polygon (Nearest Neighbor) Method
Advantages
• Easy to Use
• Quick Calculation in 2D
Disadvantages
• Discontinuous Estimates
• Edge Effects difficult to
do in 3D
2. Triangulation Method
• The technique of removing possible discontinuities between adjacent
points by fitting a plane through three samples that surround the
point being estimated. The equation of a plane can be expressed
generally as
z = ax + by + c (11.1)
2. Triangulation 140
+1
477
+2
696
+5 +6
606 791
Method *P
???
Calculation of Weights by Area 130
+3 +4 +7
1. Weight of 2 = Area 5X3 / Area 253 = 0.511 227 646 783
2. Weight of 5 = Area 2X3 / Area 253 = 0.273
60 70 80
3. Weight of 3 = Area 2X5 / Area 253 = 0.216
➔ Estimate = (0.511)(696) + (0.273)(227) +
(0.216)(606) = 548.7
2. Triangulation 140
+1
+2
696
+5 +6
606 791
Method (Delaunay Triangle) 477
*P
???
130
+3 +4 +7
227 646 783
60 70 80
2. Triangulation Method
• Triangulation (continued)
• Advantages of Method
• Easy to Understand
• Fast Calculations in 2D
• Can Be Done Manually
• Disadvantages
• Triangulation Network Is Not Unique. The Use of Delaunay Triangles Is an Effort to Work With A
“Standard” Set of Triangles
• Not Useful for Extrapolation
• Difficult to Implement in 3D
3. Local Sample Mean
• All Samples Weighted Equally Within Local Neighborhood
• If All Data Shown Are In the Local Neighborhood the Estimate = 603.7
Well X Y Value Distance to P
140 +2 +5 +6
+1 696 606 791 1 61 139 477 4,47
477
*X 2 63 140 696 3,61
???
3 64 129 227 8,06
4 68 128 646 9,49
130
+3 +4
5 71 140 606 6,71
+7
227 646 783 6 73 141 791 8,94
7 75 128 783 13,45
60 70 80
3. Local Sample Mean
• Advantages
• Easy to Understand
• Easy to Calculate in Both 2D and 3D
• Fast
• Disadvantages
• Local Neighborhood Definition is Not Unique
• Location of Samples is Not Used Except to Define Local Neighborhood
• Sensitive to Data Clustering
• Does Not Honor the Data. At Data Locations, This Method Does Not
Return the Data Value
• This Method Is Rarely Used!
4. Inverse Distance Methods
• Sample Weight is Inversely Proportional to Some Exponent of the Distance Between the
Sample and the Point Being Estimated
• The Estimate is Given By
1
( )
n
d p z ( x i )
z* = i=1 n
1
d p
i=1
• where d = distances, z(x) = sample values, p = exponent
• Note: Local Sample Mean is Equivalent to Exponent = 0 and the Nearest Neighbor Method is
Equivalent to Exponent = Infinity.
Estimation
• Inverse Distance Methods (continued)
• For Example Data, the Calculation Yields Using an Exponent of 1, 2, and 3
Yields
Exp = 1 Exp = 2
Well X Y Value Distance to X 1/d w = (1/d)/[Sum of (1/d)] w * Data Value Well X Y Value Distance to X 1/d w = (1/d)/[Sum of (1/d)] w * Data Value
1 61 139 477 4,47 0,2236 0,2098 100,0938 1 61 139 477 4.47 0.0500 0.2582 123.1502
2 63 140 696 3,61 0,2774 0,2603 181,1513 2 63 140 696 3.61 0.0769 0.3972 276.4476
3 64 129 227 8,06 0,1240 0,1164 26,4224 3 64 129 227 8.06 0.0154 0.0794 18.0326
4 68 128 646 9,49 0,1054 0,0989 63,9021 4 68 128 646 9.49 0.0111 0.0574 37.0627
5 71 140 606 6,71 0,1491 0,1399 84,7755 5 71 140 606 6.71 0.0222 0.1147 69.5356
6 73 141 791 8,94 0,1118 0,1049 82,9918 6 73 141 791 8.94 0.0125 0.0645 51.0544
7 75 128 783 13,45 0,0743 0,0698 54,6168 7 75 128 783 13.45 0.0055 0.0285 22.3373
1,0656 1,0000 593,9537 0.1937 1.0000 597.6204
Exp = 3
Well X Y Value Distance to X 1/d w = (1/d)/[Sum of (1/d)] w * Data Value
1 61 139 477 4.47 0.0112 0.2746 130.9832
2 63 140 696 3.61 0.0213 0.5240 364.7007
3 64 129 227 8.06 0.0019 0.0469 10.6389
4 68 128 646 9.49 0.0012 0.0288 18.5828
5 71 140 606 6.71 0.0033 0.0814 49.3056
6 73 141 791 8.94 0.0014 0.0343 27.1509
7 75 128 783 13.45 0.0004 0.0101 7.8974
0.0407 1.0000 609.2596
18
• Inverse Distance Methods (continued)
19
• Inverse Distance Methods
(continued)
• Comparison of Results
Obtained From Different
Exponents
Exponent = 1 Exponent = 2 Exponent = 3
20
• Advantages
• Easy to Understand
• Easy to Implement
• Changing Exponent Adds Some Flexibility to Adapt method to Different
Estimation Problems
• This Method Can Handle Anisotropy
• Disadvantages
• Difficulties Encountered When Point to Estimate Coincides With Data Point
(d = 0, Weight is Undefined)
• Possible Solutions
• Assign Data Value to Point to Be Estimated
• Add Small Constant to Weights (Data Are No Longer Honored!)
• Susceptible to Clustering
21
Local Estimation
• Limitations of Traditional Estimation Methods
• Weights Are Based On Arbitrary Schemes
• No Model of Spatial Continuity Is Used
• Estimates Are Biased Towards Clustered Data
• No Measure of Estimate Uncertainty
• Estimated Field of Values Is Much Smoother Than the
Underlying Random Field (Function) That Was
Sampled. (This Is True For All Estimation Techniques,
Including Kriging.)
22
Clustering
• Types of Areal Data
Distributions
Uniform Regular
Random
Clustered Anisotropic
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