l_read.me

it is a source code from gps toolbox

L_READ.ME
ASCII-file accompanying the MATLAB M-files for the LAMBDA-method
for GPS double difference ambiguity resolution.

Development of the LAMBDA method
--------------------------------
The LAMBDA method was introduced at the IAG General Meeting in 1993 [1]
and has from then on been further developed and implemented [2]
at the Delft Geodetic Computing Centre at the Delft University of 
Technology, the Netherlands.

[1] Teunissen P.J.G. (1993). Least-squares estimation of the integer
GPS ambiguities. Invited lecture. Section IV Theory and Methodology.
General Meeting of the International Association of Geodesy. Beijing,
China. August 1993. Also available in: publications of the Delft Geodetic
Computing Centre, LGR-series no. 6, 16 pp.

[2] Jonge P.J. de and C.C.J.M. Tiberius (1996). The LAMBDA method for
integer ambiguity estimation: implementation aspects. Publication of the
Delft Geodetic Computing Centre, LGR-series no. 12, 49 pp.

For a brief description of the method, for up-to-date information on
the implementation and a bibliography, refer to the homepage of the LGR
on Internet:  http::/www.geo.tudelft.nl/mgp/
The information can be found under 'Precise GPS positioning' and 
reference [2] is available as PostScript file.

Purpose of the method
---------------------
The LAMBDA method solves the integer minimization problem:
min_a ||\hat{a} - a||_{Q_{\hat{a}}^{-1}}^2  with a in Z^n.
Based on the real valued least-squares estimate \hat{a} for the vector of
(integer) parameters, the minimization is solved by a discrete search 
over an ellipsoidal region (in the R^n space), defined by the covariance
matrix. The gridpoint that has least distance to the real estimate
(in the metric of the covariance matrix) represents the INTEGER
LEAST SQUARES ESTIMATE \check{a}.

The acronym stands for Least squares AMBiguity Decorrelation Adjustment.
The method is used for estimating the integer GPS double difference
ambiguities.

The LAMBDA method in MATLAB
---------------------------
The LAMBDA method envisages an efficient solving of the minimization
problem, by first decorrelating the ambiguities, and a minimum storage,
by employing a so-called depth-first search.

The current implementation can be optimized, but for explanatory reasons
it was decided to stay close to the existing description.

..................

Brief outline of the method
---------------------------
The function lambda should be provided with \hat{a} and Q_{\hat{a}},
respectively the vector with estimates for the GPS double difference
ambiguities from the float solution and the corresponding variance 
covariance matrix. The output is \check{a}, the integer least squares 
estimate for the vector of ambiguities.

Prominent feature of the method is that the ambiguities are decorrelated
prior to the search over the ellipsoidal region. The decorrelation
is materialized in the Z-transformation-matrix and it maps the vector
containing the n original integer ambiguities a to a vector of new integer
ambiguities z.

The procedure to compute the integer estimate reads:
- triangular decomposition of the variance covariance matrix
- construct and apply (implicitly) the decorrelating Z-transformation
- invert the (updated) triangular decomposition
- determine an appropriate size for the search ellipsoid
- find the best candidate (gridpoint) inside the ambiguity search
  ellipsoid
- back transformation, to get the integer estimate for the vector of 
  original double difference ambiguities


December 6, 1996