Awange J.,Grafarend E., Palancz B., Zaletnyik P. Algebraic Geodesy and Geoinformatics
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374 Index
planar
conventional solution, 250
Gro ebner basis solution, 251
three-dimension
closed form, 254
conventional solution, 255
Grafarend-Shan method, 262
Gro ebner basis solution, 255
overdetermined, 259
photogrammetric, 261
resultant solution, 256
invariant, 303
inverse, 14
ionosphere, 266, 269
ionospheric errors, 278, 290
isotropic, 121
isotropic dilatation, 116
isotropic error, 135
isozenithal, 158
J. A. Grunert, 218, 226
Jacobi matrix, 101, 108, 189
Jacobian determinant, 52, 54, 177, 178,
207, 233, 257
Kronecker, 38
Kronecker-Zehfu decomposition, 134
Kummer, 38
L1-signals, 281
L2-signals, 281
Lagrangean, 123, 161, 163
centralized, 124
multiplier, 36
reduced, 124
unconstraint, 123
Leading Coefficient (LC), 340
Leading Monomial (LM), 39, 229, 340
Leading Term (LT), 340
Least Common Multiple (LCM), 41, 229
least square, 79
optimization, 84
least squares, 49, 79, 101, 111, 122, 135,
181, 199, 245, 305, 315, 318
levelling, 24, 89, 90
lexicographic ordering, 36, 40–42, 44, 164,
176, 222, 228, 230, 252, 276, 339
graded, 44, 339
graded reverse, 340
linear algebra, 14, 49
linear Gauss-Markov model, 79, 135
lo cal level system, 142
lo cal methods, 101, 329
LPS systems, 140
manifold, 88
Maple, 36
Gro ebner basis computation, 46
Mathematica, 24, 25, 27, 36
Gro ebner basis computation, 45
Matlab, 24, 25, 27, 36
“det” command, 51
“p olyfit” function, 279
“randn” command, 151
“ro ots” command, 36, 40, 51, 174, 196,
224, 232, 277
matrix properties, 117
maximum, 36
mean error matrix, 129, 132
minimal norm, 166
minimization problem, 80
minimum, 36
minimum distance mapping (MDM), 156
mo dels
Gauss-Markov, 87
linear Gauss-Markov, 90
mixed, 87
non-sto chastic, 87
stochastic, 87
monoid, 16
monomial ideal, 103
monomial orde ring, 339
monomials, 17
multideg, 40, 340
multidimensional scaling, 114
multipath, 139, 290
multiplicative associativity, 14
multiplicative commutativity, 14
multiplicative distributivity, 15
multiplicative identity, 10, 14
multiplicative inverse, 11, 14
multipolynomial resultants, 49, 51
nadir-viewing microwave, 268
necessary condition, 53, 101, 162, 180
network densification, 3
Newton polytope, 31
Newton-Raphson method, 325
Index 375
Newton-type methods, 101, 325
Niels Henrick Abel, 22
imp ossibility theorem, 22
nonlinearity, 2
nonlinearity errors, 273
NSolve, 195
numbers
complex, 9, 11, 13
invention of zero, 10
natural, 9, 10, 22
prime, 21
rational, 9, 11, 13, 15
real, 9, 11, 13, 14
ring, 22
numerical methods , 1
objective function, 80, 329
orientation unknown, 140, 144, 145
orthogonal group, 120
orthonormal matrix, 120, 126
outliers, 123, 289
overdetermined system, 338
overdetermined systems, 79, 101
P-co de, 141, 269
P. J. Huber, 4, 290
partial Procrustes algorithm, 219
PDOP, 181, 182
permutation, 93
perspective center, 244, 262
phase delay, 271
photo coordinates, 9, 13
Pizetti’s projection, 156
planar ranging, 21
polygon, 31
polynomials, 3, 9, 17, 18
S–polynomial, 41, 42, 229
critical de gree, 52
cubic, 25, 54, 230
degree zero, 223
equivalence, 40
extraneous factor, 53
factorization, 21
homogeneous, 49, 53
imp ossibility theorem, 22
irreducible, 21
Legendre, 1
minimal, 22
multivariate, 18, 19
of degree zero, 50
ordering
lexicographic, 36
quadratic, 21, 24, 54
quartic, 26, 27, 219, 256, 275, 308
reduced cubic, 25
resultants, 33, 49
ro ots, 2, 24, 25, 27, 36
solvability test, 44
Sylvester resultant, 274
total degree, 52
univariate, 3, 12, 19, 36, 163, 165, 232
polytope, 31
positional error sphere, 333
positional norm, 180
positive definite, 87, 90, 162
positive semi-definite, 123
Pothenot’s problem, 219
precipitable water, 271
pressure, 139, 173, 266, 269, 273
Procrustes
application in medicine, 114
gene recognition, 114
general solution, 116
identification of malarial parasites, 115
magic bed, 112
partial solution, 111, 116
software, 114
Procrustes algorithm, 101
projection lines, 160
Pythagorean length, 11
Pythagorean theorem, 275
quadratic formula, 24
quaternions, 9, 86
quotient, 11
radial distance, 180
radical solutions, 22
radio waves, 269
radiometers, 268
radiosondes, 265
Random Access Memory (RAM), 3
random errors, 86, 89
ranging, 173
3d-ranging
closed form, 203
elimination approach-1, 204
elimination approach-2, 205
376 Index
Gro ebner approach, 206
ovedetermined case, 211
resultant approach, 207
GPS pseudo-ranging
Gro ebner basis solution, 179
Sturmfels’ solution, 177
planar
conventional approach, 192
overdetermined, 198
reduced Groebner approach, 194
Sylvester approach, 193
reduced Groebner basis, 37, 45, 274, 276
redundant observations, 79
reference ellipsoid, 155
reference frames, 141
refraction angle, 269, 273
refractive index, 270
remote sensing, 173
resection, 217
3dimension
exact solution, 226
Grafarend et al. approach, 237
Gro ebner basis solution, 228
Grunert’s equations, 226
overdetermined, 240
photogrammetric, 244
resultant solution, 230
planar
analytic solution, 220
Gro ebner basis approach, 222
Sturmfels’ resultant approach, 223
scanner application, 217
residual, 101
residuals, 191, 290
rings, 9
commutative, 16, 19
congruence, 14
definition, 15
division, 16
fruit, 14
mo dular arithmetic, 14, 18
numbers, 9, 13, 18
polynomial, 9, 14, 16, 18, 19
with identity, 15, 16
rob otics, 1
robust estimation, 4, 290
Ro drigues, 11
ro ot-mean-square-errors, 191
ro ots , 24
rotation, 116, 129, 303
rotation matrix, 116, 119, 148, 262, 334
Runge-Kutta method, 70
satellites
CHAMP, 266
clo ck bias, 175
COSMIC, 266
ephemerids, 141
GALILEO, 111, 139, 140, 173, 267
geostationary, 265
GLONASS, 111, 139, 140, 173, 267
GNSS, 139, 173
GPS, 111, 139, 141, 173, 266
GRACE, 266
LEO, 269
o ccultations, 2, 278
polar, 265
receiver range bias, 175
remote sensing, 266
SAC-C, 266
velocities, 270, 274
scale factor, 119, 303, 334
second moments, 94
semi-group, 16
semi-major axis, 159, 168
semi-minor axis, 159, 168
semi-norm, 127, 128
set, 10, 11
shape analysis, 111
singular value decomposition (SVD), 118,
119, 123, 126, 128
singular values, 107
singularity, 102
skew-symmetric matrix, 305, 309
slant water, 272
smo oth-linear bending, 115
Snell’s law, 270
solutions
analytic, 3
closed form, 3
exact, 2
least squares, 19, 89
numerical, 3
space angles, 226, 249, 254, 303
Spatial Reference System, 173
spherical harmonics, 1
square matrix, 53
square root, 11
Index 377
standard basis, 33
standard deviation, 152
standard errors, 152
starshaped surface, 159
stencil method, 237
stochastic, 327
stochasticity, 4, 97
stratosphere, 266, 268
sufficient condition, 53, 162, 180
surface normal, 159
surjective mapping, 120
Sylvester resultants, 49, 50, 207
symbolic solution, 321
Synthetic Aperture Radar (SAR), 268
Taylor points, 94
Taylor series, 102
Taylor series expansion, 189
TEC, 267
temp erature, 139, 173, 265, 266, 269, 273,
286
Theseus, 113
top ographical surface, 159
Topography, 155
total degree, 49, 53, 231, 339, 340
transformations
7-parameter, 4, 111
affine, 115
Cartesian to ellipsoidal, 156, 167
Gauss-Jacobi solution, 313
Gro ebner solution, 305
Karussel, 143
orthogonal, 111
orthogonal Procrustes, 120
similarity, 120, 249, 262
translation, 116, 119, 129, 303, 334
trigonometric addition formulae, 274
trop opause , 268, 286
trop os phere, 268
UCAR, 266
univariate polynomial, 321
unknown orientation, 143, 147, 220, 222,
223, 251
variance-covariance, 181, 211, 333
matrix, 94
propagation, 92, 93, 95, 291, 313
vectors, 14
vertical deflection, 112
vertical deflections, 140
W-semi-norm, 121
W. R. Hamilton, 12, 86
W. WU, 61
water vapour, 173, 265, 266, 269, 273, 286
weather, 139, 265
weight matrix, 97, 291
weighted arithmetic mean, 89
Weitzenb¨ock space, 120, 249
World Geodetic System WGS-84, 140
zenith delay, 271