Helmert-Wolf blocking
Encyclopedia
The Helmert–Wolf blocking (HWB) is a least squares
solution method for a sparse canonical block-angular (CBA) system of linear equation
s. Friedrich Robert Helmert
(1843–1917) reported on the use of such systems for geodesy
in his book "Die mathematischen und physikalischen Theorieen der höheren Geodäsie, 1. Teil" published in Leipzig, 1880. Helmut Wolf (1910–1994) published his direct semianalytic solution based on ordinary Gaussian elimination
in matrix
form in his paper "The Helmert block method, its origin and development", Proceedings of the Second International Symposium on Problems Related to the Redefinition of North American Geodetic Networks, Arlington, Va. April 24–April 28, 1978, pages 319–326. The HWB solution is really fast to compute but it is optimal only if observational errors do not correlate between the data blocks. Fortunately, the generalized canonical correlation
analysis (gCCA) is the statistical method of choice for making those harmful cross-covariances vanish. This may, however, become quite tedious depending on the nature of the problem.
ing (FKF) by augmenting its linear regression
equation system to take into account information from numerical forecasts, physical constraints and other ancillary data sources that are available in realtime. Operational accuracies can then be computed reliably from the theory of minimum-norm quadratic unbiased estimation (Minque
) of C. R. Rao
(1920– ).
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
solution method for a sparse canonical block-angular (CBA) system of linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s. Friedrich Robert Helmert
Friedrich Robert Helmert
Friedrich Robert Helmert was a German geodesist and an important writer on the theory of errors.Helmert was born in Freiberg, Kingdom of Saxony. After schooling in Freiberg and Dresden, he entered the Polytechnische Schule, now Technische Universität, in Dresden to study engineering science in 1859...
(1843–1917) reported on the use of such systems for geodesy
Geodesy
Geodesy , also named geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal...
in his book "Die mathematischen und physikalischen Theorieen der höheren Geodäsie, 1. Teil" published in Leipzig, 1880. Helmut Wolf (1910–1994) published his direct semianalytic solution based on ordinary Gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
in matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
form in his paper "The Helmert block method, its origin and development", Proceedings of the Second International Symposium on Problems Related to the Redefinition of North American Geodetic Networks, Arlington, Va. April 24–April 28, 1978, pages 319–326. The HWB solution is really fast to compute but it is optimal only if observational errors do not correlate between the data blocks. Fortunately, the generalized canonical correlation
Generalized canonical correlation
In statistics, the generalized canonical correlation analysis , is a way of making sense of cross-correlation matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes Principal component analysis to two sets of random variables, a gCCA ...
analysis (gCCA) is the statistical method of choice for making those harmful cross-covariances vanish. This may, however, become quite tedious depending on the nature of the problem.
Applications
The HWB method is a "must" in Satellite Geodesy and similar large problems. The HWB method can be extended to fast Kalman filterFast Kalman filter
The fast Kalman filter , devised by Antti Lange , is an extension of the Helmert-Wolf blocking method from geodesy to real-time applications of Kalman filtering such as satellite imaging of the Earth...
ing (FKF) by augmenting its linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
equation system to take into account information from numerical forecasts, physical constraints and other ancillary data sources that are available in realtime. Operational accuracies can then be computed reliably from the theory of minimum-norm quadratic unbiased estimation (Minque
Minque
In statistics, the theory of minimum norm quadratic unbiased estimation was developed by C.R. Rao. Its application was originally to the estimation of variance components in random effects models.The theory involves three stages:...
) of C. R. Rao
C. R. Rao
Calyampudi Radhakrishna Rao FRS known as C R Rao is an Indian statistician. He is currently professor emeritus at Penn State University and Research Professor at the University at Buffalo. Rao has been honored by numerous colloquia, honorary degrees, and festschrifts and was awarded the US...
(1920– ).
External links
- http://www.ngs.noaa.gov/GRD/GPS/DOC/gpscom/mca.html GPScom Software Documentation