C10(3): The Ten Parameter Conformal Group as a Datum Transformation in Three-Dimensional Euclidean Space - Map Projections: Cartographic Information Systems - page 660

the input of several norm estimation principles, namely in order to demonstrate various statistical
properties. In particular, we have applied maximum likelihood estimations based on
• double exponential distribution ( Laplace distribution )

• normal distribution ( Gauss distribution )

• rectangular distribution

leading to
• least absolute value estimation on the basis of balanced observation (BLAVE)

• method of least squares (LESS)

• minimizing of the maximum (absolute) residual (Minimax)

as estimation principles. Tables 24.3 , 24.4 and 24.5 list the computed residuals after a C 10 (3)-
fit. Note that centered coordinates yield the same residuals as non-centered coordinates. The
maximum (absolute) value has been indicated by a boldface letter. The number of zero residuals
(consistent system) equals the number of unknown parameters in “BLAVE” ( conditio sine qua
non ). Here uniqueness of the solution may be proved by a standard simplex algorithm. Especially
all “BLAVE” zero residuals are indicated by underlining. Both the revised simplex algorithm as
a two phase method as well as iteratively reweighting of least squares were applied and result in
identical numerical values. The residuals from a Minimax fit indicate that the number of the max-
imum (absolute) residual (0.691, 462, 6) equals the number of unknown parameters plus one. The
revised simplex-algorithm with a two phase method has been computationally used. Table 24.6
reviews the balancing factors being used with BLAVE. Introducing these factors to the diagonal
matrix of a priori variances of observations yields a constant value for the diagonal elements
of the corresponding orthogonal projection. This fact insures the equal (balanced) influence of
all observations to the estimation result. Finally Tables 24.7 and 24.8 points on the estimated
parameters for LESS, BLAVE and Minimax for the reduced data as well as for the original data.

E 3 , 3D-terrestrial coordinates X T := [ X 1 ,X 2 ,X 3 ]

Table 24.1 Forty-five Cartesian coordinates of 15 points in

in NSWC SZ-2 datum ( Heindl 1986 , p. 27)

X 1 (m) X 2 (m) X 3 (m)

No. Station name

1 Kampen 3,632,765.990 532,701.670 5,198,129.630

2 Panker 3,664,812.680 682,336.390 5,158,383.280

3 Norderney 3,753,388.490 476,213.990 5,117,811.275

4 Hohenbunstorf 3,778,287.710 698,731.870 5,074,177.910

5 Damme 3,846,623.740 555,451.830 5,040,702.130

6 Koterberg 3,895,883.440 639,735.270 4,993,422.110

7 Xanten 3,940,607.900 447,253.300 4,978,805.700

8 Kloppenheim 4,041,958.920 620,750.390 4,878,752.340

9 Coburg 4,010,635.330 779,127.830 4,882,227.200

10 Furth 4,126,024.610 526,134.090 4,819,991.010

11 Wettzell 4,075,658.770 931,927.670 4,801,684.040

12 Oberkochen 4,145,376.680 737,471.190 4,776,079.930

13 Feldberg 4,245,506.720 597,066.970 4,708,719.100

14 HohenpeiBenberg 4,213,876.500 819,884.640 4,702,970.260

15 Pfander 4,254,358.420 733,416.600 4,680,984.940