Least Squared Error Data Reduction

After attending a couple of NARAM's and helping host one, I had plenty of opportunity to see how altitude data was handled. I wasn't very impressed with the add-hoc method used to combine data from more than two tracking stations so tried to think of something that might be better. What I came up with is a least squared error technique.

The method approved in the NAR Sporting Code (a.k.a. the "Pink Book") is to average together the results which close. This is a simple method but it completely ignores the quality of the different results..

The geodesic data reduction method finds the line with minimum length that is perpendicular to the vectors from the two tracking stations. The altitude of the midpoint of this line is the result and the ratio of the lines length to the altitude is the closure figure of merit. (FOM) This is usually called the error but is no such thing. It is just an easily computed metric.

I extend this by finding the point which minimizes the sum of the squared distances from a point to each of the tracking vectors. This is a Least Squared Error (LSE) method. In the two station case, this produces the exact same result as the geodesic method.

For a closure figure of merit I take a different approach. I could use a ratio of the sum of the squared errors to the altitude but I decided that using the angles between the tracking station vectors and the solution point made better sense.

One advantage of this LSE method is that I must locate each tracking station within a 3D cartesian coordinate system. It is therefore trivial to use stations at different absolute altitudes.

To test this method I decided to do two basic tests: Process old contest results to compare the two methods. And apply random error to known exact azimuth and elevation angles and look at the variance in the results.

Here is part of that first test. This table has the E Eggloft Altitude results from NARAM-44. Included are the raw tracking data. (azimuth and elevation) Note that values of '-99' indicate a track lost. (I removed all cases where three or more stations reported track lost.) Then there are the altitude (altx) and closure (clx) numbers from the geodesic method. Values of 0 and 100 indicate that one of the two stations reported a track lost. Then there is the averaged result. The results from my method give the (x,y,z) location of the solution point. I also compute the angular error between each station (alpha, bravo, charlie, delta) and this point. Any angle greater than 5 degrees indicates that this station was not used in the solution. I finish up by computing the percentage difference between the two methods.

az1 el1 az2 el2 az3 el3 az4 el4 alt 1 cl 1 alt2 cl 2 alt 3 cl 3 alt 4 cl 4 alt 5 cl 5 alt 6 cl 6 alt. x y z alpha bravo charlie delta geodesic LSE % difference
33.5 48.5 42 52.5 28 76.5 6.5 54 456.7 2.2 442.0 2.8 485.2 3.0 496.0 1.8 473.1 0.7 466.2 0.6 470 342.3 233.9 467.3 0.6 1.0 1.3 1.0 470 467 0.6
72 67.5 23 43 10.5 54 37 78.5 538.1 2.3 497.3 1.7 517.9 1.0 520.9 3.0 529.1 1.9 517.7 2.1 520 77.8 208.3 523.0 1.1 0.9 0.7 0.6 520 523 0.6
36.5 43.5 43.5 44.5 17 71.5 4 48.5 376.1 6.1 352.4 0.0 376.8 3.6 406.7 2.0 386.3 0.2 370.7 0.7 378 327.3 249.9 376.2 1.3 1.2 1.0 0.8 378 376 0.5
2 55 0.5 79 94 68 39.5 51.5 668.4 2.3 685.7 2.5 770.7 9.5 514.1 6.2 680.6 0.7 686.1 10.6 664 459.1 -7.1 675.2 1.8 0.9 2.2 4.1 664 675 1.7
14 49.5 17.5 55 60 59.5 40.5 45 399.6 0.4 387.9 1.0 431.1 12.1 321.0 4.4 402.4 2.2 393.0 16.7 378 324.3 66.4 395.4 1.7 2.3 2.2 4.8 378 395 4.5
37.5 45.5 37.5 40 23.5 59.5 17.5 49.5 346.5 10.6 295.1 7.5 335.3 2.1 323.7 2.4 340.2 1.8 328.3 14.8 324 275.1 208.5 332.8 1.6 3.6 1.3 2.2 324 333 2.8
64.5 44 23.5 18 15.5 25 50.5 55.5 196.2 19.5 150.4 7.9 182.9 3.7 184.8 1.1 179.9 0.7 171.1 26.4 174 88.6 184.5 180.1 2.7 3.4 2.1 3.7 174 180 3.4
58 10.5 -99 -99 311 21.5 329.5 15.5 0.0 100.0 0.0 100.0 94.3 4.1 97.1 3.9 96.0 6.6 0.0 100.0 96 286.2 456.2 96.4 0.4 91.7 0.7 0.3 96 96 0.0
4 56.5 -99 -99 -99 -99 47.5 45 0.0 100.0 0.0 100.0 0.0 100.0 355.7 2.7 0.0 100.0 0.0 100.0 356 240.2 16.7 355.0 0.6 132.3 128.4 0.6 356 355 0.3
-99 -99 39 7.5 334.5 13.5 283 28 0.0 100.0 104.4 13.1 107.5 1.9 0.0 100.0 0.0 100.0 100.4 6.8 104 32.6 460.7 102.9 93.6 0.6 0.8 0.4 104 103 1.0
34 56 -99 -99 -99 -99 10 59 0.0 100.0 0.0 100.0 0.0 100.0 580.0 2.5 0.0 100.0 0.0 100.0 580 331.6 220.0 578.0 0.6 142.0 160.3 0.6 580 578 0.3
25.5 39 34.5 40.5 44 55 21.5 39 285.7 11.3 249.3 9.8 273.7 3.1 284.7 2.4 279.0 1.7 275.9 11.3 272 322.8 158.9 276.4 1.6 3.4 2.3 0.8 272 276 1.5
31.5 47 37.5 49 37 63 24 50.5 402.2 3.7 347.0 12.2 391.8 2.7 338.1 3.3 369.4 4.5 390.9 17.2 375 306.8 175.5 381.4 1.2 4.3 1.9 3.5 375 381 1.6
15.5 45 -99 -99 84.5 69.5 30.5 43.5 0.0 100.0 0.0 100.0 482.1 15.0 347.8 3.4 452.6 1.1 0.0 100.0 400 419.3 94.7 439.1 2.0 149.1 3.8 4.4 400 439 9.8
26 54.5 -99 -99 70 78 -99 -99 0.0 100.0 0.0 100.0 0.0 100.0 0.0 100.0 606.3 5.5 0.0 100.0 606 399.9 176.4 606.0 1.3 148.9 1.6 141.2 606 606 0.0
5.5 57.5 6.5 70.5 75.5 67 32.5 53.5 606.3 2.1 612.9 2.1 620.3 1.2 617.6 0.1 614.0 1.2 613.3 2.5 614 388.5 32.8 612.3 0.4 0.9 0.5 0.4 614 612 0.3
40 63 30.5 55.5 23 68.5 23 67 608.9 2.0 563.4 2.8 602.5 1.2 559.9 0.1 593.5 1.8 594.3 5.5 587 237.0 191.2 593.9 0.5 1.5 0.5 1.3 587 594 1.2
8 53.5 -99 -99 130 72 17 52 0.0 100.0 0.0 100.0 838.4 2.0 888.6 1.5 859.1 1.3 0.0 100.0 862 634.6 88.8 854.0 0.4 165.1 0.6 0.6 862 854 0.9
40 65 36.5 62.5 18 78 6.5 70.5 770.9 0.9 730.8 1.7 817.6 2.6 822.1 0.9 793.5 1.6 783.1 0.6 786 279.5 237.6 784.3 0.2 0.6 1.2 0.7 786 784 0.3
358 61.5 359.5 52.5 54.5 52 48 49.5 456.5 1.2 441.4 0.1 442.8 0.1 453.8 3.9 459.2 1.5 441.8 0.0 449 254.6 -6.4 449.2 1.1 0.4 0.3 0.8 449 449 0.0
33 56.5 -99 -99 -99 -99 22 60 0.0 100.0 0.0 100.0 0.0 100.0 482.1 1.8 0.0 100.0 0.0 100.0 482 262.3 172.8 481.2 0.4 136.1 149.4 0.4 482 481 0.2
350.5 61.5 354.5 41.5 49.5 41.5 -99 -99 360.2 1.9 363.0 1.2 0.0 100.0 0.0 100.0 362.4 1.1 0.0 100.0 362 194.3 -35.9 361.1 0.5 0.4 0.2 126.5 362 361 0.3
19.5 39 29.5 44.5 53.5 55.5 25.5 38 290.3 8.1 271.2 4.5 290.2 0.1 291.9 0.2 290.2 0.5 286.9 8.4 287 340.0 126.7 287.6 0.9 2.4 0.8 0.9 287 288 0.3
30 40.5 32.5 41 36.5 60.5 17 42.5 301.2 2.8 292.2 1.8 302.5 1.7 309.2 1.9 307.1 1.2 297.6 0.2 302 312.1 181.0 301.2 0.6 0.7 0.8 0.4 302 301 0.3
-99 -99 37 62.5 13.5 78 -99 -99 0.0 100.0 755.7 1.6 0.0 100.0 0.0 100.0 0.0 100.0 0.0 100.0 756 289.6 238.7 754.5 146.3 0.4 0.4 154.6 756 754 0.3
48 53.5 43 47 5 67.5 11 61.5 507.8 7.4 425.0 7.3 477.8 6.8 419.8 2.7 489.1 2.2 465.5 16.0 464 255.5 262.7 474.5 1.8 3.8 0.3 3.7 464 474 2.2
34 42 28.5 38 28.5 59 15.5 45.5 292.2 0.9 299.8 9.7 307.4 2.3 315.3 1.6 311.0 1.2 293.3 7.9 303 280.2 189.4 300.7 0.4 2.1 1.9 1.4 303 301 0.7
55.5 65 17.5 37.5 19.5 47 44.5 67 397.2 0.7 388.4 0.5 384.6 1.3 397.2 3.8 395.1 1.1 388.3 1.2 392 111.3 152.6 391.3 1.0 0.1 0.3 1.1 392 391 0.3
321.5 54 329 46 66.5 40.5 64 40 428.8 4.1 437.4 6.1 458.9 0.2 444.8 1.1 451.8 0.6 437.7 4.8 443 250.3 -204.7 441.6 0.5 1.6 0.9 0.5 443 442 0.2
7 63 26 82.5 96.5 76 24 61 958.3 0.4 941.9 0.2 944.6 2.2 1003.0 0.4 949.6 0.9 960.9 1.6 960 485.5 58.9 955.1 0.1 0.2 0.5 0.6 960 955 0.5
26.5 64 53 74 58.5 83.5 11.5 66 981.0 1.1 914.0 4.8 915.1 0.8 927.0 0.2 908.9 0.3 965.5 2.6 935 413.6 203.1 942.8 0.2 1.5 1.2 0.3 935 943 0.9
40 66.5 90 50.5 25 58.5 46 66.5 593.4 79.0 132.3 184.4 526.3 5.2 460.1 2.7 510.8 0.2 375.3 132.2 499 164.0 132.4 506.7 1.0 -45.7 1.1 2.0 499 507 1.6
44.5 20 -99 -99 244 43 334.5 25.5 0.0 100.0 0.0 100.0 309.6 4.5 305.3 5.7 291.8 2.3 0.0 100.0 302 578.8 574.6 303.0 0.5 108.9 0.7 0.8 302 303 0.3
13.5 45 9.5 33 44 35.5 44.5 37 240.2 2.0 231.3 6.1 232.3 3.2 237.4 0.5 230.8 1.1 239.3 3.1 235 229.5 56.2 235.4 0.2 1.0 1.1 0.2 235 235 0.0
16.5 46.5 15 37 47.5 44.5 35.5 44.5 273.3 7.1 252.3 2.6 295.8 13.3 320.2 5.4 282.4 4.1 284.8 4.0 283 255.7 85.1 282.3 1.3 1.9 2.9 3.1 283 282 0.4
11.5 58.5 11.5 59.5 55.5 60.5 36 53 507.8 0.5 490.1 2.8 487.1 2.1 495.6 1.5 490.8 2.2 500.1 0.2 495 304.2 57.2 496.5 0.6 0.6 1.0 0.3 495 496 0.2
341 69.5 352 43.5 47 46 74 51 428.3 2.5 449.8 7.0 507.8 5.8 393.0 10.6 459.3 1.1 456.4 3.7 460 138.9 -63.4 451.5 2.6 0.8 2.0 3.2 460 452 1.7
17 53 60.5 75.5 117 78.5 12.5 52.5 704.9 0.5 696.5 1.5 689.6 1.5 747.2 2.3 721.6 0.2 691.8 2.7 709 515.0 161.6 702.2 0.6 0.6 0.3 0.9 709 702 1.0
12 68 9.5 59 48 63 46 62.5 605.3 1.5 577.0 0.9 646.2 0.4 599.7 2.4 616.1 1.5 614.4 2.7 610 238.6 50.5 612.6 0.3 0.8 1.0 1.0 610 613 0.5
21 35.5 55.5 56.5 76 67.5 25.5 35.5 352.6 3.8 316.5 18.2 338.6 18.1 253.5 2.3 319.4 1.7 318.4 33.2 308 405.4 135.3 312.0 2.2 -12.3 4.5 4.4 308 312 1.3
48.5 68 35.5 55.5 12.5 68 22.5 78.5 718.3 9.2 590.9 7.8 779.4 1.0 689.9 8.3 687.7 1.1 738.3 12.6 693 174.6 231.5 711.3 1.7 3.9 1.5 3.5 693 711 2.6
16.5 72 -99 -99 45 66.5 41.5 66 0.0 100.0 0.0 100.0 708.1 1.2 712.7 2.6 725.7 2.2 0.0 100.0 716 234.9 62.9 716.7 0.9 148.8 0.5 0.6 716 717 0.1
10.5 60.5 -99 -99 115 79 17 60.5 0.0 100.0 0.0 100.0 987.5 1.1 1084.4 1.1 982.4 1.8 0.0 100.0 1018 543.2 113.0 993.6 0.7 164.0 0.3 0.6 1018 994 2.4
346 79 352.5 65 51 66.5 59.5 68.5 918.4 1.4 910.7 4.3 958.7 0.1 937.5 2.0 985.0 1.4 905.3 2.6 936 186.7 -45.9 932.0 0.7 1.2 1.1 0.5 936 932 0.4
9.5 41.5 12.5 54 69 54 30 36.5 328.0 2.9 332.0 2.8 317.2 0.2 330.5 4.5 330.9 1.3 319.1 5.4 326 372.0 59.7 324.8 0.8 1.4 0.3 1.1 326 325 0.3
27.5 53 33 55 -99 -99 29 53.5 475.7 3.1 0.0 100.0 0.0 100.0 393.0 1.6 0.0 100.0 455.8 14.8 434 310.1 149.8 456.1 1.0 3.2 149.5 3.7 434 456 5.1
37.5 58.5 48.5 60 18 77.5 18.5 62.5 684.6 5.5 586.4 8.0 633.9 9.7 515.6 0.4 643.1 0.7 617.0 17.1 613 323.8 229.7 628.2 1.4 4.3 0.8 4.5 613 628 2.4
34.5 58.5 36.5 59.5 27 74.5 -99 -99 611.1 0.3 596.7 1.8 0.0 100.0 0.0 100.0 598.2 0.6 0.0 100.0 602 306.7 208.3 603.6 0.2 0.5 0.6 148.0 602 604 0.3
49.5 69.5 46.5 72.5 2 82.5 356.5 75.5 1283.3 5.4 1256.2 3.4 1161.8 2.0 1032.2 1.0 1113.1 4.7 1277.5 0.7 1187 305.4 297.3 1204.2 2.1 1.6 1.3 0.0 1187 1204 1.4
29.5 22.5 43.5 28.5 43.5 56 24 33 172.8 5.0 163.3 3.5 243.2 14.2 154.7 35.0 173.1 3.0 196.8 53.7 170 370.5 208.5 170.5 0.7 1.0 1.5 -14.3 170 171 0.6
58.5 68.5 43.5 62.5 348.5 76.5 338.5 82 1024.7 2.1 916.7 2.2 1240.2 0.4 891.2 8.5 996.8 1.3 1113.8 7.1 1031 187.5 343.9 1047.6 1.4 1.7 1.2 3.2 1031 1048 1.6
85.5 72 24 44.5 15.5 53.5 76 78 630.1 8.5 458.1 8.0 593.3 0.8 504.1 2.5 576.8 7.2 553.5 16.6 552 40.7 182.2 565.2 2.5 4.3 2.2 3.2 552 565 2.4
35 64 43.5 63.5 26 77 14 67 780.3 6.1 660.6 6.8 733.0 0.1 729.7 0.9 741.4 0.5 739.6 7.7 731 308.2 217.3 740.1 1.0 3.0 1.2 1.1 731 740 1.2
22.5 61.5 14.5 44 37.5 52 40 55 398.1 5.0 375.6 0.5 392.7 2.3 419.9 3.7 406.3 0.6 387.3 0.2 397 208.4 92.6 395.1 1.7 1.0 0.7 1.0 397 395 0.5
-99 -99 5 66.5 45 70 -99 -99 0.0 100.0 901.1 2.3 0.0 100.0 0.0 100.0 0.0 100.0 0.0 100.0 901 218.2 38.9 898.7 161.2 0.6 0.6 152.5 901 899 0.2
16.5 57.5 98.5 77 141 78 22 56.5 1023.9 4.7 840.2 4.4 960.1 11.6 662.4 0.5 986.8 2.3 851.3 17.8 878 586.9 151.2 901.7 1.8 4.2 1.5 4.8 878 902 2.7
320.5 65.5 332 64 68 55 62 54.5 759.5 5.9 686.3 2.4 759.2 1.6 821.6 0.3 767.0 0.3 749.9 3.1 757 268.5 -199.0 752.4 1.3 1.8 0.7 0.2 757 752 0.7
349.5 34 343 45.5 74.5 34.5 55 26 249.2 1.7 250.6 1.1 297.8 15.1 178.1 14.4 250.5 0.5 268.7 26.5 250 365.0 -69.5 249.7 0.3 0.4 0.1 -9.3 250 250 0.0
6 24.5 7 26.5 57 25 50.5 20.5 143.3 1.5 142.0 8.6 159.9 12.2 109.9 20.1 137.3 0.6 152.2 39.8 141 303.5 33.8 140.7 0.4 1.2 1.0 -10.4 141 141 0.0
26.5 9.5 19 5.5 37.5 8.5 33 8 41.4 22.1 37.4 8.7 40.2 4.8 42.9 11.5 42.4 19.0 38.0 3.4 39 245.2 119.7 40.3 1.2 0.7 0.8 0.2 39 40 2.6
25.5 71 15 58.5 39 65.5 39.5 67.5 662.6 1.0 617.4 0.8 673.2 2.3 670.0 1.2 670.9 2.9 653.4 0.0 658 213.0 97.6 659.5 0.6 0.6 1.2 0.3 658 659 0.2
75.5 78 33.5 61.5 12 72 8 87 1153.4 9.3 783.4 5.1 1196.6 6.1 1265.8 0.5 1134.7 4.4 1109.3 7.5 1107 80.0 252.8 1098.1 1.7 3.6 3.0 1.5 1107 1098 0.8
3.5 70.5 21 84.5 94.5 77.5 27.5 66.5 1347.5 1.3 1149.6 1.5 1226.2 2.0 1262.6 3.5 1303.9 2.8 1241.3 0.5 1255 477.2 28.0 1265.7 1.2 0.8 1.0 0.7 1255 1266 0.9
21 73 20.5 70 49.5 76.5 29.5 71 954.1 1.8 870.3 2.4 995.3 1.3 997.9 1.6 1009.9 1.9 932.0 0.2 960 285.9 111.8 957.1 0.8 0.9 1.3 0.3 960 957 0.3
-99 -99 341 69.5 73.5 67 -99 -99 0.0 100.0 818.0 8.5 0.0 100.0 0.0 100.0 0.0 100.0 0.0 100.0 818 337.9 -77.0 817.0 156.0 2.3 2.2 141.5 818 817 0.1
32.5 58.5 22.5 52 33.5 65 -99 -99 482.1 2.9 486.0 3.7 0.0 100.0 0.0 100.0 493.0 2.8 0.0 100.0 487 254.3 152.1 487.0 0.9 0.9 1.0 144.0 487 487 0.0
14 30 23 30.5 53.5 35.5 32.5 25.5 182.7 21.6 160.1 10.2 174.5 0.0 183.3 5.4 181.6 0.6 174.1 20.2 180 316.7 90.9 176.6 2.5 4.4 1.4 1.3 180 177 1.7
21.5 32.5 11 14.5 37.5 13.5 45 17 116.9 13.4 90.2 28.8 81.6 8.5 106.1 41.3 96.6 21.6 94.9 13.9 82 189.0 73.8 98.3 6.7 1.4 2.9 3.6 82 98 19.5
41.5 61.5 38 58 12 75.5 13 66 665.0 2.3 633.1 0.7 658.4 4.6 600.9 2.1 680.3 0.5 632.9 5.0 645 277.7 235.3 650.8 0.9 1.1 1.0 1.8 645 651 0.9
18 53.5 19 52 50 60 31 49.5 414.5 2.5 398.9 1.2 411.5 0.2 418.7 2.8 422.4 1.1 404.4 0.5 412 298.8 98.7 411.5 0.9 0.7 0.6 0.5 412 412 0.0
314 24 339 15.5 61.5 12 -99 -99 115.4 18.7 103.3 5.6 0.0 100.0 0.0 100.0 125.2 9.8 0.0 100.0 114 196.0 -179.0 117.3 3.3 3.0 2.4 94.7 114 117 2.6
9 50.5 -99 -99 49.5 42.5 43 40.5 0.0 100.0 0.0 100.0 289.7 2.4 302.0 4.5 298.2 3.2 0.0 100.0 297 251.0 38.6 295.5 1.2 127.3 0.7 0.9 297 296 0.3
84 71 27.5 42.5 10.5 53 74.5 81.5 597.8 15.0 434.4 8.6 581.1 0.7 550.8 0.1 574.7 1.6 537.3 18.3 535 37.2 212.6 554.5 2.6 5.0 1.7 3.0 535 554 3.6
8 71.5 12 61.5 48.5 63.5 42 64.5 689.8 6.6 590.5 3.6 659.0 5.1 758.0 2.8 684.0 1.8 665.2 1.8 674 237.1 49.6 669.8 1.9 2.0 1.7 1.1 674 670 0.6
7.5 62.5 23.5 74.5 75 72.5 30 59.5 773.2 4.3 703.4 2.9 766.2 1.0 783.2 0.0 765.2 0.2 760.5 3.7 759 397.0 61.6 760.4 0.7 1.7 0.7 0.3 759 760 0.1
353.5 62.5 342.5 77.5 90.5 68.5 -99 -99 820.5 0.8 795.6 5.3 0.0 100.0 0.0 100.0 886.7 1.4 0.0 100.0 834 440.6 -50.2 838.2 0.4 1.4 1.6 139.9 834 838 0.5
81 75.5 24.5 58.5 0.5 70.5 13.5 87.5 999.7 0.1 952.8 5.3 1144.4 0.7 1011.9 0.2 1107.8 1.8 1005.5 0.3 1037 46.6 262.7 1037.2 0.3 1.1 1.7 0.4 1037 1037 0.0
25 57 72 73.5 137.5 84 9 60.5 878.5 0.7 826.4 0.9 942.8 1.7 785.4 2.8 867.3 1.5 885.8 5.2 864 510.9 229.8 875.8 0.6 1.1 0.6 1.6 864 876 1.4
15.5 37 122.5 62 163 62 6.5 37 578.4 8.8 510.3 6.2 559.9 0.5 589.0 2.3 586.2 0.1 549.1 7.1 562 741.8 217.4 561.2 1.2 3.1 0.8 0.3 562 561 0.2
34 72.5 36.5 67 30.5 75 22.5 74 978.5 7.5 758.2 8.0 913.1 2.6 970.9 0.3 912.0 1.0 930.8 7.4 911 253.6 180.7 918.6 1.3 3.4 1.8 0.7 911 919 0.9
29.5 61.5 31.5 62.5 40 76 17 64 652.5 0.4 631.9 2.6 689.7 1.7 684.3 0.7 675.0 1.4 660.6 0.9 666 313.9 178.7 663.9 0.1 0.7 1.1 0.6 666 664 0.3
19 76 3 47 36 54 59.5 64 511.6 3.2 528.6 7.2 543.8 0.6 543.7 2.0 551.2 1.6 516.1 6.2 533 130.1 40.2 528.5 0.6 1.9 1.3 1.1 533 528 0.9
5.5 65 2 42 44 44.5 58 50 379.9 0.6 368.8 0.2 385.7 0.4 362.0 0.3 379.6 3.2 377.4 2.6 376 177.6 11.1 377.8 0.8 0.4 0.7 0.8 376 378 0.5
43 67.5 29.5 58.5 19.5 74 12 72.5 715.3 1.8 662.8 4.7 782.3 3.0 796.4 0.2 771.3 2.3 719.4 1.8 741 227.9 215.4 735.6 0.6 1.3 1.9 0.9 741 736 0.7
25.5 62 21 55 38.5 64.5 -99 -99 527.6 2.1 508.4 0.0 0.0 100.0 0.0 100.0 532.2 0.5 0.0 100.0 523 256.0 124.5 526.0 0.5 0.7 0.4 144.2 523 526 0.6
31 67 23 59 32 70.5 35 66.5 652.2 1.6 638.3 2.9 686.2 6.1 589.5 0.5 683.6 0.1 637.4 5.7 648 241.7 140.8 653.2 0.4 1.0 1.6 2.3 648 653 0.8
21 29.5 21 27 50 39.5 39.5 27 171.9 5.8 167.0 12.6 202.5 21.5 145.4 6.2 184.5 5.7 168.5 34.2 167 304.4 113.9 175.4 1.2 2.1 2.4 -9.6 167 175 4.8
358 48.5 358 51 64.5 46 42.5 38.5 353.1 0.3 336.1 1.4 337.2 1.7 347.4 4.7 348.6 3.9 340.5 0.1 344 318.0 -11.9 344.4 1.2 0.3 0.8 0.7 344 344 0.0
34 62.5 28 55 30 68 33 63.5 569.6 3.6 527.2 0.6 587.0 6.8 490.0 1.1 575.4 1.3 547.5 11.0 550 246.7 158.6 557.9 0.6 2.1 1.2 3.3 550 558 1.5
34 69.5 -99 -99 31 59 47.5 66.5 0.0 100.0 0.0 100.0 561.0 1.1 516.1 1.2 545.7 1.9 0.0 100.0 541 168.5 105.6 545.8 0.8 136.8 0.5 0.9 541 546 0.9
7.5 70.5 12.5 66.5 54 67.5 41 65 768.2 4.8 670.1 2.9 760.2 2.8 791.9 0.7 760.0 1.8 750.3 3.3 750 272.8 45.2 751.7 1.0 1.8 1.3 0.3 750 752 0.3
16.5 71 10 65.5 50.5 70 37.5 66.5 767.2 1.9 742.1 3.1 786.6 1.6 765.7 1.8 789.8 3.2 758.0 2.4 768 264.4 68.0 769.0 0.9 0.9 1.1 0.5 768 769 0.1
11 53.5 46 76.5 98.5 73.5 21 52 667.5 3.1 626.1 1.7 660.5 1.2 683.9 0.6 666.8 0.4 658.1 1.9 660 485.0 100.6 659.9 0.6 1.1 0.6 0.1 660 660 0.0
24 66.5 22 52 37 59.5 41.5 61.5 530.0 11.3 443.2 6.1 517.6 0.5 514.0 1.4 524.2 0.6 500.4 11.7 500 215.7 107.2 509.5 2.1 3.9 1.1 1.6 500 509 1.8
42 53.5 -99 -99 -99 -99 341 60.5 0.0 100.0 0.0 100.0 0.0 100.0 931.9 2.5 0.0 100.0 0.0 100.0 932 497.0 457.3 926.7 0.6 144.2 177.5 0.6 932 927 0.5
42 71 32.5 66.5 19.5 78.5 8.5 75 960.5 0.3 904.5 1.5 990.0 2.8 1029.7 0.6 984.8 1.9 961.1 1.6 972 253.0 224.8 967.6 0.3 0.5 1.1 0.8 972 968 0.4
41 77.5 10 49 29.5 57 70.5 69.5 584.5 0.5 560.9 3.5 654.3 3.1 539.3 3.6 611.8 2.6 589.4 5.1 590 96.1 78.7 595.6 0.8 0.9 1.7 2.3 590 596 1.0
27 58 -99 -99 49 79 -99 -99 0.0 100.0 0.0 100.0 0.0 100.0 0.0 100.0 660.1 0.3 0.0 100.0 660 368.1 186.1 659.8 0.1 148.6 0.1 145.5 660 660 0.0
33 80 18 63 28 68.5 56.5 77 938.6 6.1 831.6 3.4 954.8 0.7 940.3 1.7 932.9 1.3 924.3 6.4 920 135.4 107.5 925.8 1.1 2.2 0.4 1.2 920 926 0.7
37 64 17 43.5 27.5 53 41.5 63 428.4 1.4 419.4 0.1 432.6 1.0 434.2 0.5 429.4 0.4 428.5 0.8 429 167.2 127.7 428.7 0.2 0.4 0.4 0.2 429 429 0.0
13.5 56.5 16.5 61.5 60 64.5 30.5 53 513.7 0.0 496.4 0.1 509.4 2.8 524.3 1.0 511.3 2.2 510.8 1.3 511 333.6 78.3 510.4 0.4 0.1 0.9 0.6 511 510 0.2

That is a lot of data to look at!

(The curious can read the C source.)

Surprisingly, a large number of the angles are around 1 degree or less. Given the nature of the equipment used, this really amazes me. The people working the tracking stations at NARAM are good!

For the most part, the difference between the two methods is slight. So it is hard to say from this which method is better. The Monte-Carlo simulation results will be needed to differentiate the two methods. But by looking at the times the two methods disagree, we still might learn something.

Here is one track where the difference is almost 10%.

az1 el1 az2 el2 az3 el3 az4 el4 alt 1 cl 1 alt2 cl 2 alt 3 cl 3 alt 4 cl 4 alt 5 cl 5 alt 6 cl 6 alt. x y z alpha bravo charlie delta geodesic LSE % difference
15.5 45 -99 -99 84.5 69.5 30.5 43.5 0.0 100.0 0.0 100.0 482.1 15.0 347.8 3.4 452.6 1.1 0.0 100.0 400 419.3 94.7 439.1 2.0 149.1 3.8 4.4 400 439 9.8

Station 2 reported 'track lost' so there are only three possible geodesic solutions. Of these only two close at 347.8m (3.4%) and 452.6m (1.1%). Both solutions indicate that they closed but they produce wildly different results. The LSE results show that all three of the valid stations were kept in the solution with angles of 2.0, 3.8, and 4.4 degrees. I had the program set with a limit of 5 degrees before throwing out a station. If I drop that down to 4 degrees, the station 4 drops out. This results in angles for the two remaining stations of less than 1 degree. Station 4 ends up with and angle of 7 degrees so it looks it was pointing off in the wrong direction.

This shows that the geodesic method can produce results that look good but are obviously not. Where the LSE method can recognize this problem and take care of it. At least when a criteria of 4 degrees error or less is used.

Here is another track where the difference is almost 20%.

az1 el1 az2 el2 az3 el3 az4 el4 alt 1 cl 1 alt2 cl 2 alt 3 cl 3 alt 4 cl 4 alt 5 cl 5 alt 6 cl 6 alt. x y z alpha bravo charlie delta geodesic LSE % difference
21.5 32.5 11 14.5 37.5 13.5 45 17 116.9 13.4 90.2 28.8 81.6 8.5 106.1 41.3 96.6 21.6 94.9 13.9 82 189.0 73.8 98.3 6.7 1.4 2.9 3.6 82 98 19.5

All four stations reported but only one of the six possible geodesic solutions closed. And with only a closure figure of 8.5% at that. The LSE method used three of the four stations to arrive at its solution.

Part of the problem here is the very low altitude. The final altitude is under 100 meters and the baselines are all much longer than that. So the distance from the tracking stations was relatively large. The length of the line between the tracking vectors depends on the angular error and the distance from the stations to the track point. This varies only slightly with altitude. But since the geodesic closure figure is the ratio of this length to altitude, low flights are penalized with poor closure figures.

This case has a difference of 5.1%.

az1 el1 az2 el2 az3 el3 az4 el4 alt 1 cl 1 alt2 cl 2 alt 3 cl 3 alt 4 cl 4 alt 5 cl 5 alt 6 cl 6 alt. x y z alpha bravo charlie delta geodesic LSE % difference
27.5 53 33 55 -99 -99 29 53.5 475.7 3.1 0.0 100.0 0.0 100.0 393.0 1.6 0.0 100.0 455.8 14.8 434 310.1 149.8 456.1 1.0 3.2 149.5 3.7 434 456 5.1

One station reported track lost leaving three possible geodesic solutions. Only two of these closed. The two closed values were 475m (3.1%) and 393m (1.6%). Both closure figures indicate high confidence in the solutions. yet they differ by almost 80 m. They both can't be correct. The LSE method used all three stations in its solution with angles of 1.0, 3.2, and 3.7 degrees. So this is a case where the trackers didn't quite agree on where the rocket was resulting in higher than typical variance in the tracking angles. At least with the LSE method you can see this. As to which of the two methods produces the best result it is hard to say. I have my biased opinion but a definitive answer will have to await the Monte-Carlo runs.

One of the interesting side benefits of the LSE method is that it provides a 3D solution rather than just an altitude. This lets you plot the location of all of the flights.

For the curious, the tracking stations at NARAM-44 had the following (x,y) positions:

Station X (meters) Y (meters)
1 (Alpha) 0.0 0.0
1 (Bravo) 598.0 0.0
1 (Charlie) 449.0 285.0
1 (Delta) -4.0 267.0

One of the really bizarre results I found in the NARAM 46 data was the following:

az1 el1 az2 el2 az3 el3 alt 1 cl 1 alt2 cl 2 alt 3 cl 3 x y z alpha bravo charlie geodesic LSE % difference
35 67 -28 71 25 51.5 332 48 497 4.5 302 0.16 105 -73 302 0.05 25.75 0.03 400 302 24

Only three tracking stations arranged in a triangle were used at NARAM 46. The fourth station was not setup because a suitable location at the same elevation as the other three could not be found. Positive angles were to the inside of the triangle so obviously station 2 is pointing outside of that triangle. In spite of this obvious problem, Contest Manager reports a closed track using it. The result is an altitude that is way off. If you look at the angular error from the LSE solution, two stations are spot on while station 2 is off by 25 degrees. If you go with the assumption that the station 2 azimuth should have positive 28 degrees instead you do get a track not closed result but now the other station does close with station 2 but is once again very different at 453 meters.

Monte Carlo runs

In order to better compare the results of the various data reduction methods, I wanted to see how they performed in the presence of random noise. Since I don;t have a good feel for the probability distribution function of real data, I will simply use nice zero mean Gaussian noise. The standard distribution for this noise is 1 degree which seems to be about the right amount to simulate the magnitude of typical errors.

So. pick a point in space and then compute the azimuth and elevation angles for each tracking station. Now added random noise to each of these and round to the nearest half degree (simulating the typical resolution of tracking hardware) and compute the results. Repeat.

After a few thousand runs, compute the standard deviation of the percentage error. To get a feel for things, I wrote the program to use a fixed altitude but to cover the area around the tracking stations with a grid. I then plotted the results using Gnuplot. This was interesting if not very informative until I figured out how to add contours. For now all I have is code to do this for the two station case. Getting all of the angles right to keep the geodesic method happy is giving me fits for more than two stations at the moment. For the two station case, LSE and geodesic (and John DeMar's HIT method as well) produce essentially the same results. So I will put only one plot here:

In this plot, the tracking stations are at x=0, y=0 and x=1000, y=0. Target altitude was 2000. While the error doesn't change much over a very large volume, there is still a definite "sweet spot" on either side of the baseline.

Here are the error plots using the tracking station setup we had for NARAM-44. These are run for altitudes of 300, 600, and 1200 meters. I am putting them side by side to make comparison easier. But if you are stuck with a narrow monitor it might not work too well. Least squared error method on the left and averaged geodesic on the right.

While the least squared error method does have distinctly lower variance than the averaged geodesic method, it isn't a big advantage.

That was the standard deviation of the error, here are some plots of the the average error. The noise in the input data is zero mean. It would be nice if the resulting altitude errors were zero mean but because the equations are highly non-linear, it isn't very likely.

The geodesic method appears to be biased slightly high and the least squared method is biased slightly low. The magnitudes of the bias are similar.

From this, it appears that the performance of the averaged geodesic method is quite good and the least squared method doesn't provide much of an improvement.

In analyzing the NARAM-44 data, I looked at the angle errors for the least squared solution. I broke these down into azimuth and elevation errors. The azimuth error standard deviation was about twice that for the elevation. So what happens to the Monte-Carlo results if the azimuth noise is bumped to a standard deviation of 2 degrees? Once again, least squares is on the left and geodesic on the right.

That was interesting. I had to restrict the range for the geodesic data because it broke down at the edges of the test region and produced very large numbers for the average error and standard deviation. This also messed up the contours. I would have to dig into the results to be sure (but I have no desire to dig into thousands of lines of numbers) but this is probably due to none of the possible solutions closing. The least squared method remains well behaved and again provides better performance than the geodesic method.

Just for grins, I ran the program again but with the four stations placed how it used to be done: in pairs at the end of a 600 meter baseline. Here are the results for a target altitude of 600 meters.

Which isn't very different. The advantage in putting the tracking stations in a rectangular layout is in increasing the chances of at least two trackers seeing the rocket. If you have two trackers at one spot, if one has trouble with the sun, so will the other.

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