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Kriging for Drop Test Data

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What is the coverage of these retardant drops?

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We use the drop test to measure coverage.

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Retardant is collected in cups. Cups are capped and weighed.

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Knowing the weights allows us to recreate the pattern as a contour plot.

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Each cup becomes a gpc value, grams per hundred square feet.

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Converting grams of retardant into gpc.

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Both plots below use the same raw data. In the first plot, the contours are created from an array of raw data at a 15 by 15 spacing and by interpolation routines inherent in Splus software. In the second plot the points between raw data are estimated using kriging to produce an array of higher resolution (7.5 by 7.5) and then by Splus software. 15 by 15 array 7.5 by 7.5 array Which one is more accurate?

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We use a weighted average of the raw gpc values to predict unknown values. We use a method called kriging to come up with weights for the weighted average. These kriging weights are based on distance, direction and variability of the raw data.

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Steps to find the weighted average: 1.Create a variogram which is a mathematical description of the variation between observed gpc values. 2.Use the variogram to estimate the covariance at different distances. 3.Use the covariances to calculate kriging weights. 4.Calculate the weighted average of the raw data.

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Step 1: Create a variogram which is a mathematical description of the variation between observed gpc values. The variogram is the average squared difference between pairs of gpc values. The table on the next slide shows the variogram values for drop 104.

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distancegamma (variogram)number of pairs 23.970960.1458764120 58.11540.29422718028 86.777920.33411911831 111.96150.48188430293 145.63260.55369829346 170.10470.61312534471 201.50290.67990742091 227.9750.69438341259 257.24550.78118850939 283.37950.77115942779 310.97560.87573561842 342.3950.88940960465 369.55410.92866261478 400.31730.97599267232 427.09810.98746560637 456.36251.04414272292 484.2881.04921963810 512.66380.94881675918 542.54890.91198468710 569.68040.80848473019 In the first line of the table we see that at a distance of 23.97096 feet there are 4,120 pairs of cups. The average squared difference between all these pairs is 0.145876. Note: The distances in this table are calculated from the length of the drop pattern and the grid spacing.

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We plot the data from the table and fit a curve. distance gamma (variogram) 23.970960.145876 58.11540.294227 86.777920.334119 111.96150.481884 145.63260.553698 170.10470.613125 201.50290.679907 227.9750.694383 257.24550.781188 283.37950.771159 310.97560.875735 342.3950.889409 369.55410.928662 400.31730.975992 427.09810.987465 456.36251.044142 484.2881.049219 512.66380.948816 542.54890.911984 569.68040.808484

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Range: As the distance between pairs increases, the average squared difference between pairs will increase. Eventually, an increase in distance will no longer produce and increase in average squared difference, and the variogram will reach a plateau. The distance at which the plateau is reached is the range. Sill: The value of the average squared difference at the range. Nugget: The vertical jump between zero and the first variogram value. Now that we have our range, sill and nugget we will look at some sample data to calculate the weights. Three features of the trend line are used to calculate weights in our weighted average.

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Example of gpc values in their position on the grid with unknown values we want to predict.

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For the purpose of this explanation we will only focus on 12 observed gpc values to predict one unknown value. The picture to the right shows the area we will focus on inside the box.

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Step 2: Use the variogram to estimate the covariance at different distances. Below is a table showing the position of the known values, the corresponding gpc values and the distance to the unknown values. XYZDistance from unknown value in feet 16151650.468742 23.72 26151800.937485 10.61 36151952.432393 10.61 46301651.292209 23.72 56301800.937485 10.61 66301952.280368 10.61 76451651.241534 31.82 86451801.494908 23.72 96451953.483896 23.72 106601650.68411 43.73 116601801.228865 38.24 126601953.800614 38.24

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Next, determine all the distances between known and unknown values, shown in this table. 0 unknown value 123456789101112 00.0023.7210.61 23.7210.61 31.8223.72 43.4738.24 123.720.0015.0030.0015.0021.2133.543.0033.5446.2345.0046.2354.08 210.6115.000.0015.0021.2115.0021.2133.5430.0033.5447.4345.0047.43 310.6130.0015.000.0033.5421.2115.0042.4333.5430.0054.0847.4345.00 423.7215.0021.2133.540.0015.0030.0015.0021.2133.5430.0033.5442.43 510.6121.2115.0021.2115.000.0015.0021.2115.0021.2133.5430.0033.54 610.6133.5421.2115.0030.0015.000.0033.5421.2115.0042.4333.5430.00 731.8230.0033.5442.4315.0021.2133.540.0015.0030.0015.0021.2133.54 823.7233.5430.0033.5421.2115.0021.2115.000.0015.0033.5421.2115.00 923.7242.4333.5430.0033.5421.2115.0030.0015.000.0033.5421.2115.00 1043.4745.0047.4354.0830.0033.5442.4315.0033.54 0.0015.0030.00 1138.2447.4345.0047.4333.5430.0033.5421.21 15.000.0015.00 1238.2454.0847.4345.0042.4333.5430.0033.5415.00 30.0015.000.00

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We will use a range = 430.1663057, nugget = 0.1224344, sill = 0.8386289 from the variogram we created earlier. We calculate all the covariances at those distances from the table in the previous slide. Cov(h) means the covariance at distance h

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Looking at the first column from the previous table we get: C(00.00) =sill = 0.8386289 C(23.72) = C(10.61) = 0.665113126 C(23.72) =0.606991467 C(10.61) =0.665113126 C(10.61) =0.665113126 C(31.82) =0.573660412 C(23.72) =0.606991467 C(23.72) =0.606991467 C(43.47) = 0.528895018 C(38.24) = 0.548542205 C(38.24) =0.548542205

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0123456789101112 00.83860.60700.6651 0.60700.6651 0.57370.6070 0.52890.5485 10.60700.83860.64510.58100.64510.61770.56680.70140.56680.51880.52330.51880.4912 20.66510.64510.83860.64510.61770.64510.61770.56680.58100.56680.51450.52330.5145 30.66510.58100.64510.83860.56680.61770.64510.53270.56680.58100.49120.51450.5233 40.60700.64510.61770.56680.83860.64510.58100.64510.61770.56680.58100.56680.5327 50.66510.61770.64510.61770.64510.83860.64510.61770.64510.61770.56680.58100.5668 60.66510.56680.61770.64510.58100.64510.83860.56680.61770.64510.53270.56680.5810 70.57370.58100.56680.53270.64510.61770.56680.83860.64510.58100.64510.61770.5668 80.60700.56680.58100.56680.61770.64510.61770.64510.83860.64510.56680.61770.6451 90.60700.53270.56680.58100.56680.61770.64510.58100.64510.83860.56680.61770.6451 100.52890.52330.51450.49120.58100.56680.53270.64510.5668 0.83860.64510.5810 110.54850.51450.52330.51450.56680.58100.56680.6177 0.64510.83860.6451 120.54850.49120.51450.52330.53270.56680.58100.56680.6451 0.58100.64510.8386 Our new table of covariances…

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Step 3. Use the covariances to calculate kriging weights. 0.6070 0.6651 0.6070 0.6651 0.5737 0.6070 0.5289 0.5485 1 We’ll call this matrix D = First well separate the covariance associated with the unknown value. This is in the first column or first row.

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0.60700.6651 0.60700.6651 0.57370.6070 0.52890.5485 0.83860.64510.58100.64510.61770.56680.70140.56680.51880.52330.51880.49121 0.64510.83860.64510.61770.64510.61770.56680.58100.56680.51450.52330.51451 0.58100.64510.83860.56680.61770.64510.53270.56680.58100.49120.51450.52331 0.64510.61770.56680.83860.64510.58100.64510.61770.56680.58100.56680.53271 0.61770.64510.61770.64510.83860.64510.61770.64510.61770.56680.58100.56681 0.61770.64510.58100.64510.83860.56680.61770.64510.53270.56680.58101 0.56680.53270.64510.61770.56680.83860.64510.58100.64510.61770.56681 0.58100.56680.61770.64510.61770.64510.83860.64510.56680.61770.64511 0.53270.56680.58100.56680.61770.64510.58100.64510.83860.56680.61770.64511 0.52330.51450.49120.58100.56680.53270.64510.5668 0.83860.64510.58101 0.51450.52330.51450.56680.58100.56680.6177 0.64510.83860.64511 0.49120.51450.52330.53270.56680.58100.56680.6451 0.58100.64510.83861 1111111111110 The remaining matrix we’ll call C =

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The equation to determine kriging weights is: C * kriging weights matrix = D where we want the kriging weights matrix. The easiest way to solve this equation is by using linear algebra. We can multiply both sides by C inverse… = C -1 * C * weights = D * C -1 = I * weights = D * C -1 = weights = D * C -1

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4.083-1.216-0.461-0.722-0.294-0.137-2.6550.4580.3250.5230.161-0.0640.139 -1.3044.506-1.239-0.657-0.842-0.4780.660-0.267-0.176-0.232-0.0310.0610.091 -0.441-1.2553.914-0.132-0.461-1.2280.286-0.075-0.394-0.045-0.015-0.1540.165 -1.280-0.507-0.0664.425-0.806-0.045-0.297-0.695-0.087-0.658-0.1600.1750.061 -0.452-0.802-0.441-0.8194.971-0.781-0.066-0.862-0.425-0.258-0.1700.105-0.023 -0.076-0.500-1.234-0.074-0.7904.5570.053-0.375-1.097-0.039-0.126-0.2990.054 -0.359-0.0820.039-1.066-0.3340.0164.924-1.246-0.044-1.465-0.5520.1700.053 -0.099-0.091-0.014-0.534-0.804-0.363-1.1425.029-0.7480.411-0.366-1.2790.010 0.048-0.087-0.3640.006-0.393-1.093-0.047-0.7354.548-0.200-0.570-1.1150.051 -0.186-0.0100.032-0.461-0.186-0.023-1.2980.402-0.2213.875-1.359-0.5650.162 0.0070.0190.001-0.111-0.153-0.123-0.543-0.361-0.571-1.3514.415-1.2290.080 0.0580.023-0.1670.1450.093-0.3020.126-1.274-1.110-0.560-1.2274.1950.158 0.1690.0850.1600.074-0.0200.051-0.0450.0320.0630.1900.0870.153-0.599 C -1 =

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Kriging weights = 0.0580 0.1989 0.2249 0.0384 0.1745 0.1984 0.0043 0.0378 0.0542 0.0045 0.0029 0.0034

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Notice the greater the distance the smaller the weight. XYZ Distance from unknown value (feet) WeightsProduct 16151650.468742 23.720.0580.027164915 26151800.937485 10.610.1990.186443159 36151952.432393 10.610.2250.546960529 46301651.292209 23.720.0380.049666153 56301800.937485 10.610.1750.163602356 66301952.280368 10.610.1980.452333773 76451651.241534 31.820.0040.005331766 86451801.494908 23.720.0380.056440308 96451953.483896 23.720.0540.188875433 106601650.68411 53.050.0040.003056074 116601801.228865 38.240.0030.003582303 126601953.800614 38.240.0030.012744684 1.696201453 Step 4. Calculate the weighted average of the raw data to find the weighted average.

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1.696 gpc

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Model the variation across the raw data. Use the features of the model to calculate kriging weights. Calculate the weighted average of the raw data. Re-cap:

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