This is a documentation version of GNU Gama 1.16.
Copyright (C) 2003, 2013 Aleš Čepek.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.1 or
any later version published by the Free Software Foundation; with no
Invariant Sections, with no Front-Cover Texts, and with no Back-Cover
Texts. A copy of the license is included in the section entitled "GNU
Free Documentation License".
GNU Gama 1.16
1 Introduction
1.1 Download
1.2 Install
1.2.1 Precompiled binaries for Windows
1.3 Program 'gama-local'
1.3.1 Reductions of horizontal and zenith angles
1.4 Reporting bugs
1.5 Contributors
2 XML input data format for 'gama-local'
2.1 Angular units
2.2 Prologue
2.3 Tags '' and ''
2.4 Network description
2.5 Network parameters
2.6 Points and observations
2.7 Points
2.8 Set of observations
2.9 Directions
2.10 Horizontal distances
2.11 Angles
2.12 Slope distances
2.13 Zenith angles
2.14 Azimuths
2.15 Height differences
2.16 Control coordinates
2.17 Coordinate differences (vectors)
2.18 Example of local geodetic network
3 SQL schema, SQLite and 'gama-local'
3.1 Working with SQLite database
3.2 Units in SQL tables
3.3 Network SQL definition
3.4 Table 'points'
3.5 Table 'clusters'
3.6 Table 'covmat'
3.7 Table 'obs'
3.8 Table 'coordinates'
3.9 Table 'vectors'
3.10 Example of local geodetic network in SQL
4 Network adjustment with 'gama-local'
4.1 Approximate coordinates
4.2 Gross absolute terms
4.3 Parameters of statistical analysis
4.4 Test on the reference standard deviation
4.5 Information on points
4.6 Adjusted observations and residuals
4.7 Identification of weak network elements
4.8 Estimation of real errors
4.9 Test on linearization
5 Data structures and algorithms
5.1 Observation data and points
5.2 Supported ellipsoids
5.3 Transformation from spatial to geographical coordinates
5.4 Class 'g3::Model'
Appendix A Copying This Manual
A.1 GNU Free Documentation License
A.1.1 ADDENDUM: How to use this License for your documents
Concept Index
GNU Gama 1.16
*************
1 Introduction
**************
GNU Gama is a project dedicated to adjustment of geodetic networks. It
is intended for use with traditional geodetic surveyings which are still
used and needed in special measurements (e.g., underground or high
precision engineering measurements) where the Global Positioning System
(GPS) cannot be used.
In general, surveying is the technique and science of accurately
determining the terrestrial or three-dimensional spatial position of
points and the distances and angles between them.(1)
Adjustment is a technical term traditionally used by geodesists and
surveyors which simply means "application of the least squares method to
process the over-determined system of measurements" (statistical methods
other than least squares are used sometimes but are not common). In
other words, we have more observations than needed and we are trying to
get the best estimate for adjusted observations and/or coordinates.
"Adjustment of geodetic networks" means that we have a set of points
with given coordinates coordinates of some points and a set of
observations among them. What is typical of adjustment of special
geodetic measurements is that the resulting linearised system might be
singular (we can have a network with no fixed points) and we are not
only interested in the values of 'adjusted parameters and observations'
but also in the estimates of their covariances. This is what Gama does.
Gama was originally inspired by Fortran system Geodet/PC (1990)
designed by Frantisek Charamza. The GNU Gama project started at the
department of mapping and cartography, faculty of Civil Engineering,
Czech Technical University in Prague (CTU) about 1998 and its name is an
acronym for _geodesy and mapping_. It was presented to a wider public
for the first time at FIG Working Week 2000 in Prague and then at FIG
Workshop and Seminar at HUT Helsinki in 2001.
The GNU Gama home page is
and the project is hosted on
GNU Gama is released under the GNU General Public License and is
based on a C++ library of geodetic classes and functions and a small C++
template matrix library 'matvec'. For parsing XML documents GNU Gama
calls the 'expat' parser version 1.1, written by James Clark. The
'expat' parser is not part of the GNU Gama project, and is simply used
by GNU Gama.
Adjustment in local Cartesian coordinate systems is fully supported
by a command-line program 'gama-local' that adjusts geodetic (free)
networks of observed distances, directions, angles, height differences,
3D vectors and observed coordinates (coordinates with given
variance-covariance matrix). Adjustment in global coordinate systems is
supported only partly as a 'gama-g3' program.
---------- Footnotes ----------
(1) Wikipedia,
1.1 Download
============
GNU Gama can be found in the subdirectory '/gnu/gama/' on your favourite
GNU mirror (http://www.gnu.org/prep/ftp.html) or checked-out from the
GIT. See our project page at savannah
(http://savannah.gnu.org/projects/gama/) for more information.
To get an anonymous read-only access to the GIT repository for the
latest GNU Gama source, issue the following command
git clone git://git.sv.gnu.org/gama.git
The collection of sample networks is available separetely. To checkout
the 'gama-local' examples from GIT use the command
git clone git://git.sv.gnu.org/gama/examples.git
1.2 Install
===========
GNU Gama is developed and tested under GNU/Linux. A static library
'libgama.lib' and executables are build in folders 'lib' and 'bin'. You
can compile Gama easily yourself if you download the sources. If
'expat' XML parser is installed on your system, change to the directory
of Gama project and issue the following commands at the shell prompt
$ ./configure
$ make
To run tests from the Gama test suite try
$ make check
If the script 'configure' is not available (which is the case when you
download source codes from a git server), you have to generate it using
auxiliary script 'autogen.sh'. To compile and build all binaries. Run
$ ./configure [--bindir=DIR --infodir=DIR]
$ make install
if you want also to install the binaries. You can use configure
parameters if you need to change directories where user executables and
info documentation should be installed.
Typically, if you want to download (*note Download::) and compile
sources, you will run following commands:
$ git clone git://git.sv.gnu.org/gama.git gama
$ cd gama
$ ./autogen.sh
$ ./configure
$ make
You should have 'expat' XML parser and SQLite library already
installed on your system. For example to be able to compile Gama on
Ubuntu 10.04 you have to install following packages:
make doxygen git automake autoconf libexpat1-dev libsqlite3-dev
To compile user documentation in various formats (PDF, HTML, ...) run
the following commands (before you have to run at least './configure').
$ cd doc/
$ make download-gendocs.sh
$ make run-gendocs.sh
The documentation should be in 'doc/manual' directory. To compile
API documentation run
$ doxygen
in your 'gama' directory. Doxygen output will be in the 'doxygen'
directory.
1.2.1 Precompiled binaries for Windows
--------------------------------------
GNU Gama builds for Windows are available from
1.3 Program 'gama-local'
========================
Program 'gama-local' is a simple command line tool for adjustment of
geodetic _free networks._ It is available for GNU Linux (the main
platform on which project GNU Gama is being developed) or for MS Windows
(tested with Borland compiler from Borland free command line tools and
with Microsoft Visual C++ compiler; support for Windows platform is
currently limited to maintaing compatibility with the two mentioned
compilers).
Program 'gama-local' reads input data in XML format (*note XML input
data format for gama-local::) and prints adjustment results into ASCII
text file. If output file name is not given, input file name with
extension'.txt' is used. If development files for Sqlite3 (package
'libsqlite3-dev') are installed during the build, 'gama-local' also
supports reading adjustment input data from an sqlite3 database. If run
without arguments 'gama-local' prints a short help
$ ./gama-local
Adjustment of local geodetic network version: 1.16 / GNU g++
************************************
http://www.gnu.org/software/gama/
Usage: gama-local input.xml [options]
gama-local input.xml --sqlitedb sqlite.db --configuration name [options]
gama-local --sqlitedb sqlite.db --configuration name [options]
gama-local --sqlitedb sqlite.db --readonly-configuration name [options]
Options:
--algorithm svd | gso | cholesky | envelope
--language en | ca | cz | du | es | fi | fr | hu | ru | ua | zh
--encoding utf-8 | iso-8859-2 | iso-8859-2-flat | cp-1250 | cp-1251
--angles 400 | 360
--latitude
--ellipsoid
--text adjustment_results.txt
--html adjustment_results.html
--xml adjustment_results.xml
--svg network_configuration.svg
--cov-band covariance matrix of adjusted parameters in XML output
n = -1 for full covariance matrix (implicit value)
n >= 0 covariances are computed only for bandwidth n
--iterations maximum number of iterations allowed in the linearized
least squares algorithm (implicit value is 5)
--version
--help
Program 'gama-local' version is followed by information on compiler used
to build the program (apart from GNU 'g++' compiler, two other
possibilities are 'bcc' and 'msc' for Borland and Microsoft compilers
respectively, when build under Microsoft Windows).
Option '--algorithm' enables to select numerical method used for
solution of the adjustment. Implicitly is used Singular Value
Decomposition ('svd'), alternatively user can decide for block matrix
algorithm GSO by Frantisek Charamza, based on Gram-Schmidt
orthogonalization. In both these cases, project equations are solved
directly without forming _normal equations_. Third possibility is to
select Cholesky decomposition of semidefinite matrix of normal equations
('cholesky').
Option '--language' selects language used in output protocol. For
example, if run with option '--language cz', 'gama-local' prints output
results in Czech languague using UTF-8 encoding. Implicit value is 'en'
for output in English.
Option '--encoding' enables to change inplicit UTF-8 output encoding
to iso-8859-2 (latin-2), iso-8859-2-flat (latin-2 without diacritics),
cp-1250 (MS-EE encoding) cp-12251 (Russian encoding).
Option '--angles' selects angular units to be used in output.
Options '--latitude' and/or '--ellipsoid' are used when observed
vertical and/or zenith angles need to be transformed into the projection
plane. If none of these two options is explicitly used, no corrections
are added to horizontal and/or zenith angles. If only one of these
options is used, then implicit value for '--latitude' is 45 degrees (50
gons) and implicit ellipsoid is WGS84. Mathematical formulas for the
corrections is given in the following section.
Adjustment results ('--text' and '--xml') can be redirected to
standard output if instead of a file name is used "-" string. If no
output is given, XML adjustment format is implicitly send to standard
output.
Option '--cov-band' is used to reduce the number of computed
covariances (cofactors) in XML adjustment output. Implicitly full
matrix is written to XML output, which could degrade time efficiency for
the 'envelope' algorithm for sparse matrix solution. Explicit option
for full covariance matrix is '--cov-band -1', option '--cov-band 0'
means that only a diagonal of covariance matrix is written to XML
output, '--cov-band 1' results in computing the main diagonal and first
codiagonal etc. If higher rank is specified then available, it is
reduced do maximum possible value 'dim-1'.
Option '--iterations' enables to set maximum number of iterations
allowed in the linearized least squares algorithm. After the adjustment
'gama-local' computes differences between adjusted observations computed
from residuals and from adjusted coordinates. If the positional
difference is higher than 0.5mm, approximate coordinates of adjusted
points are updated and the whole adjustment is repeated in a new
iteration. Implicit number of iterations is 5.
1.3.1 Reductions of horizontal and zenith angles
------------------------------------------------
For evaluating of reductions of horizontal and zenith angles,
'gama-local' computes a helper point P_1 in the center of the network.
Horizontal and zenith angles observed at point P_2 are transformed to
the projection plane perpendicular to the normal z_1 of the helper point
P_1. Coordinates (x_2, y_2) of point P_2 are conserved, but its normal
z_2 is rotated by the central angle 2\gamma_{12} to be parallel with
z_1.
Formulas for reductions of horizontal and zenith angles are given
only in the printed version.
1.4 Reporting bugs
==================
Undoubtedly there are numerous bugs remaining, both in the C++ source
code and in the documentation. If you find a bug in either, please send
a bug report to
bug-gama@gnu.org (mailto:bug-gama@gnu.org)
We will try to be as quick as possible in fixing the bugs and
redistributing the fixes. If you prefere, you can always write directly
to Aleš Čepek (mailto:cepek@gnu.org).
1.5 Contributors
================
The following persons (in chronological order) have made contributions
to GNU Gama project: Aleš Čepek, Jiří Veselý, Petr Doubrava, Jan
Pytel, Chuck Ghilani, Dan Haggman, Mauri Väisänen, John Dedrum, Jim
Sutherland, Zoltan Faludi, Diego Berge, Boris Pihtin, Stéphane
Kaloustian, Siki Zoltan, Anton Horpynich, Claudio Fontana, Bronislav
Koska, Martin Beckett, Jiří Novák, Václav Petráš, Jokin Zurutuza,
项维 (Vim Xiang) and Tomáš Kubín.
Václav Petráš is the author of *note SQL schema SQLite and
gama-local::.
2 XML input data format for 'gama-local'
****************************************
The input data format for a local geodetic network adjustment (program
'gama-local') is defined in accordance with the definition of Extended
Markup Language (XML) for description of structured data. The XML
definition can be found at
Input data (points, observations and other related information) are
described using XML start-end pair tags '' and '' and
empty-element tags ''.
The syntax of XML 'gama-local' input format is described in XML
schema (XSD), the file 'gama-local.xsd' is a part of the 'GNU gama'
distribution and can formally be validated independently on the program
'gama-local', namely in unit testing we use 'xmllint' validating parser,
if it is installed.
For parsing the XML input data, 'gama-local' uses the XML parser
'Expat' copyrighted by James Clark which is described at
'Expat' is subject to the Mozilla Public License (MPL), or may
alternatively be used under the GNU General Public License (GPL)
instead.
In the 'gama-local' XML input, distances are given in meters, angular
values in centigrades and their standard deviations (rms errors) in
millimeters or centigrade seconds, respectively. Alternatively angular
values in 'gama-local' XML input can be given in degrees and seconds
(*note Angular units::). At the end of this chapter an example of the
'gama-local' XML input data object is given.
2.1 Angular units
=================
Horizontal angles, directions and zenith angles in 'gama-local' XML
adjustment input are implicitly given in gons and their standard
deviations and/or variances in centicentigons. Gon, also called
centesimal grade and Neugrad (German for new grad), is 1/400-th of the
circumference. For example
The same angular value (direction) can be expressed in degrees as
In XML adjustment input degrees are coded as a single string, where
degrees (57), minutes (32) and seconds (28.428) are separated by dashes
(-) with optional leading sign. Spaces are not allowed inside the
string. Gons and degrees may be mixed in a single XML document but one
should be careful to supply the information on standard deviations
and/or covariances in the proper corresponding units.
Internally 'gama-local' works with gons but output can be transformed to
degrees using the option '--angles 360'.
2.2 Prologue
============
XML documents begin with an XML declaration that specifies the version
of XML being used (_prolog_). In the case of 'gama-local' follows the
root tag '' with XML Schema namespace defined in attribute
'xmlns':
GNU Gama uses non-validating parser and the XML Schema Definition
namespace is not used in 'gama-local' but it is essential for usage in
third party software that might need XML validation.
2.3 Tags '' and ''
=======================================
A pair tag '' contains a single pair tag '' that
contains the network definition. The definition of the network is
composed of three sections:
* '' of the network (annotation or comments),
* network '' and
* '' section.
The sections '' and '' are optional, the
section '' is mandatory. These three sections may
be presented in any order and may be repeated several times (in such a
case, the corresponding sections are linked together by the software).
The pair tag '' has two optional attributes 'axes-xy' and
'angles'. These attributes are used to describe orientation of the 'xy'
orthogonal coordinate system axes and the orientation of the observed
angles and/or directions.
* 'axes-xy="ne"' orientation of axes 'x' and 'y'; value 'ne' implies
that axis 'x' is oriented north and axis 'y' is oriented east.
Acceptable values are 'ne', 'sw', 'es', 'wn' for left-handed
coordinate systems and 'en', 'nw', 'se', 'ws' for right-handed
coordinate systems (default value is 'ne').
* 'angles="right-handed"' defines counterclockwise observed angles
and/or directions, value 'left-handed' defines clockwise observed
angles and/or directions (default value is 'left-handed').
Many geodetic systems are right handed with 'x' axis oriented east, 'y'
axis oriented north and counterclockwise angular observations. Example
of left-handed orthogonal system with different axes orientation is
coordinate system _Krovak_ used in the Czech Republic where the axes 'x'
and 'y' are oriented south and west respectively.
GNU Gama can adjust any combination of coordinate and angular systems.
Example
=======
...
...
It is planned in future versions of the program to allow more
'' tags (analysis of deformations etc.) and definitions of new
tags.
2.4 Network description
=======================
The description of a geodetic network is enclosed in the start-end pair
tags ''. Text of the description is copied into the
adjustment output and serves for easier identification of results. The
text is not interpreted by the program, but it may be helpful for users.
Example
=======
A short description of a geodetic network ...
2.5 Network parameters
======================
The network parameters may be listed with the following optional
attributes of an empty-element tag ''
* 'sigma-apr = "10"' value of a priori reference standard
deviation--square root of reference variance (default value 10)
* 'conf-pr = "0.95"' confidence probability used in statistical tests
(dafault value 0.95)
* 'tol-abs = "1000"' tolerance for identification of gross absolute
terms in project equations (default value 1000 mm)
* 'sigma-act = "aposteriori"' actual type of reference standard
deviation use in statistical tests ('aposteriori | apriori');
default value is 'aposteriori'
* 'update-constrained-coordinates = "no"' enables user to control if
coordinates of constrained points are updated in iterative
adjustment. If test on linerarization fails (*note
Linearization::), Gama tries to improve approximate coordinates of
adjusted points and repeats the whole adjustment. Coordinates of
constrained points are implicitly not changed during iterations.
* 'algorithm = "gso"' numerical algortihm used in the adjistment
(gso, svd, cholesky, envelope).
* 'languade = "en"' the language to be used in adjustment output.
* 'encoding = "utf-8"' adjustment output encoding.
* 'angles = "400"' output results angular units (400/360).
* 'latitude = "50"'
* 'ellipsoid'
* 'cov-band = "-1"' the bandwith of covariance matrix of the adjusted
parameters in the output XML file (-1 means all covariances).
Values of the attributes must be given either in the double-quotes
('"..."') or in the single quotes (''...''). There can be _white
spaces_ (spaces, tabs and new-line characters) between attribute names,
values, and the _equal_ sign.
Example
=======
2.6 Points and observations
===========================
The points and observations section is bounded by the pair tag
'' and contains information about points, observed
horizontal directions, angles, and horizontal distances, height
differences, slope distances, zenith angles, observed vectors and
control coordinates.
Optional attributes of the start tag '' allow for
the definition of default values of standard deviations corresponding to
observed directions, angles, and distances.
* 'direction-stdev = "..."' defines the implicit value of standard
deviation of observed directions (default value is not defined)
* 'angle-stdev = "..."' defines the implicit value of standard
deviation of observed angles (default value is not defined)
* 'zenith-angle-stdev = "..."' defines the implicit value of standard
deviation of observed zenith angles (default value is not defined)
* 'azimuth-stdev = "..."' defines the implicit value of standard
deviation of observed azimuth angles (default value is not defined)
* 'distance-stdev = "..."' defines the implicit value of standard
deviation of observed distances, horizontal or slope (default value
is not defined)
Implicit values of standard deviations for the observed distances are
calculated from the model with three constants _a_, _b_, and _c_
according to the formula
a + bD^c,
where _a_ is a constant part of the model and _D_ is the observed
distance in kilometres. If the constants _b_ and/or _c_ are not given,
default values _b=0_ and _c=1_ will be used.
Example
=======
2.7 Points
==========
Points are described by the empty-element tags '' with the
following attributes:
* 'id = "..."' is the point identification attribute (mandatory);
point identification is not limited to _numbers_; all printable
characters can be used in identification.
* 'x = "..."' specifies coordinate 'x'
* 'y = "..."' specifies coordinate 'y'
* 'z = "..."' specifies coordinate 'z', point height
* 'fix = "..."' specifies coordinates that are fixed in adjustment;
acceptable values are 'xy', 'XY', 'z', 'Z', 'xyz', 'XYZ', 'xyZ' and
'XYz'.
* 'adj = "..."' specifies coordinates to be adjusted (unknown
parameters in adjustment); acceptable values are 'xy', 'XY', 'z',
'Z', 'xyz', 'XYZ', 'xyZ' and 'XYz'.
With exception of the first attribute (point id), all other attributes
are optional. Decimal numbers can be used as needed.
Control coordinates marked using the 'fix' parameter are not changed in
the adjustment. Uppercase and lowercase notation of coordinates with
the 'fix' parameter are interpreted the same. Corrections are applied
to the unknown parameters identified by coordinates written in lowercase
characters given in the 'adj' parameter. When the coordinates are
written using uppercase, they are interpreted as _constrained
coordinates._ If coordinates are marked with both the 'fix' and 'adj',
the 'fix' parameter will take precedence.
_Constrained coordinates_ are used for the regularization of free
networks. If the network is not free (fixed network), the _constrained_
coordinates are interpreted as other unknown parameters. In classical
free networks, the _constrained_ points define the regularization
constraint
\sum dx^2_i+dy^2_i = \min.
where _dx_ and _dy_ are adjusted coordinate corrections and the
summation index _i_ goes over all _constrained_ points. In other words,
the set of the _constrained_ points defines the adjustment of the free
network (its shape and size) with a simultaneous transformation to the
approximate coordinates of selected points. Program 'gama-local' allows
the definition of constrained coordinates with 1D leveling networks, 2D
and 3D local networks.
Example
=======
2.8 Set of observations
=======================
The pair tag '' groups together a set of observations which are
somehow related. A typical example is a set of directions and distances
observed from one stand-point. An observation section contains a set of
* horizontal directions ''
* horizontal distances ''
* horizontal angles ''
* slope distances ''
* zenith angles ''
* azimuths ''
The band variance-covariance matrix of directions, distances, angles or
other observations listed in one '' section may be supplied using a
'' pair tag with attributes 'dim' (dimension) and 'band'
(bandwidth). The band-width of the diagonal matrix is equal to 0 and a
fully-populated variance-covariance matrix has a bandwidth of 'dim-1'.
Observation variances and covariances (i.e. an upper-symmetric part of
the band-matrix) are written row by row between '' and
'' tags. If present, the dimension of the variance-covariance
matrix must agree with the number of observations.
The following example of variance-covariance matrix with dimension 6 and
bandwidth 2 (two nonzero codiagonals and three zero codiagonals)
[ 1.1 0.1 0.2 0 0 0
0.1 1.2 0.3 0.4 0 0
0.2 0.3 1.3 0.5 0.6 0
0 0.4 0.5 1.4 0.7 0.8
0 0 0.6 0.7 1.5 0.9
0 0 0 0.8 0.9 1.6 ]
is coded in XML as
1.1 0.1 0.2
1.2 0.3 0.4
1.3 0.5 0.6
1.4 0.7 0.8
1.5 0.9
1.6
If two or more sets of directions with different orientations are
observed from a stand-point, they must be placed in different ''
sections. The value of an orientation angle can be explicitly stated
with an attribute 'orientation="..."'. Normally, it is more convenient
to let the program calculate approximate values of orientations needed
for the adjustment. If directions are present, then the attribute
'station' must be defined.
Optional attribute 'from_dh="..."' enables to enter implicit height of
instrument for all observations within the '' pair tag.
Observed distances are expressed in meters, their standard deviations in
millimeters. Observed directions and angles are expressed in
centigrades (400) and their standard deviations in centigrade seconds.
Height differences can be entered in the '' or
'' section. If entered in the '' section, the
'dist="..."' parameter is ignored (*note Height differences::).
Example
=======
100.00 100.00 100.00 25.00
2.9 Directions
==============
Directions are expressed with the following attributes in an
empty-element tag ''
* 'to = "..."' target point identification
* 'val = "..."' observed direction; *note Angular units::
* 'stdev = "..."' standard deviation (optional)
* 'from_dh = "..."' instrument height (optional)
* 'to_dh = "..."' reflector/target height (optional)
The standard deviation is an optional attribute. However since all
observations in the adjustment must have their weights defined, the
standard deviation must be given either explicitly with the attribute
'stdev="..."' or implicitly with '' or with a variance-covariance matrix for the
given observation set. A similar approach applies to all the
observations (distances, angles, etc.)
Example
=======
2.10 Horizontal distances
=========================
Distances are written using an empty-element tag '' with
attributes
* 'from = "..."' standpoint identification
* 'to = "..."' target identification
* 'val = "..."' observed horizontal distance
* 'stdev = "..."' standard deviation of observed horizontal distance
(optional)
* 'from_dh = "..."' instrument height (optional)
* 'to_dh = "..."' reflector/target height (optional)
Contrary to directions, distances in an observation set ('') do not
need to share a common stand-point. An example is set of distances
observed from several stand-points with a common variance-covariance
matrix.
Example
=======
2.11 Angles
===========
Observed angles are expressed with the following attributes of an
empty-element tag ''
* 'from = "..."' standpoint identification (optional)
* 'bs = "..."' backsight target identification
* 'fs = "..."' foresight target identification
* 'val = "..."' observed angle; *note Angular units::
* 'stdev = "..." ' standard deviation (optional)
* 'from_dh = "..."' instrument height (optional)
* 'bs_dh = "..."' backsight reflector/target height (optional)
* 'fs_dh = "..."' foresight reflector/target height (optional)
Similar to distance observations, one observation set may group angles
observed from several standpoints.
Example
=======
2.12 Slope distances
====================
Slope distances (space distances) are written using an empty-element tag
'' with attributes
* 'from = "..."' standpoint identification (optional)
* 'to = "..."' target identification
* 'val = "..."' observed slope distance
* 'stdev = "..."' standard deviation of observed slope distance
(optional)
* 'from_dh = "..."' instrument height (optional)
* 'to_dh = "..."' reflector/target height (optional)
Similar to horizontal distances, one observation set may group slope
distances observed from several standpoints.
Example
=======
2.13 Zenith angles
==================
Zenith angles are written using an empty-element tag '' with
the following attributes
* 'from = "..."' standpoint identification (optional)
* 'to = "..."' target identification
* 'val = "..."' observed zenith angle; *note Angular units::
* 'stdev = "..."' standard deviation of observed zenith angle
(optional)
* 'from_dh = "..."' instrument height (optional)
* 'to_dh = "..."' reflector/target height (optional)
Similar to horizontal distances, one observation set may group zenith
angles observed from several standpoints.
Example
=======
2.14 Azimuths
=============
The azimuth is defined in GNU Gama as an observed horizontal angle
measured from the North to the given target. The true north orientation
is measured by gyrotheodolites, mainly in mine surveying. In Gama
azimuths' angle can be measured clockwise or counterclocwise according
to the angle orientation defined in '' tag.
Azimuths are expressed with the following attributes in an empty-element
tag ''
* 'from = "..."' standpoint identification
* 'to = "..."' target point identification
* 'val = "..."' observed azimuth; *note Angular units::
* 'stdev = "..."' standard deviation (optional)
* 'from_dh = "..."' instrument height (optional)
* 'to_dh = "..."' reflector/target height (optional)
The standard deviation is an optional attribute. However since all
observations in the adjustment must have their weights defined, the
standard deviation must be given either explicitly with the attribute
'stdev="..."' or implicitly with '' or with a variance-covariance matrix for the
given observation set.
Example
=======
2.15 Height differences
=======================
A set of observed leveling height differences is described using the
start-end tag '' without parameters. The
'' tag can contain a series of height differences
(at least one) and can optionally be supplied with a variance-covariance
matrix. Single height differences are defined with empty tags ''
having the following attributes:
* 'from = "..."' standpoint identification
* 'to = "..."' target identification
* 'val = "..."' observed leveling height difference
* 'stdev = "..."' standard deviation of levellin elevation and
* 'dist = "..."' distance of leveling section (in kilometers)
If the value of standard deviation is not present and length of leveling
section (in kilometres) is defined, the value of standard deviation is
computed from the formula
m_dh = m_0 sqrt(D_km)
If the value of standard deviation of the height difference is defined,
information on leveling section length is ignored. A third possibility
is to define a common variance-covariance matrix for all elevations in
the set.
Example
=======
2.16 Control coordinates
========================
Control (known) coordinates are described by the start-end pair tag
''. A series of points with known coordinates can be
defined using the '' tag. The variance-covariance matrix for
the entire set of points can be created with a single '' tag.
In the '' tags, a point identification (ID) and its coordinates
(x, y and z) must be listed. Although the order of the '' tag
attributes is irrelevant in the corresponding variance-covariance
matrix, the expected order of the coordinates is x, y and z (the
horizontal coordinates x, y, or the height z might be missing, but not
both). The type of the points may be defined either directly within the
'' tag or outside of it.
Example
=======
...
2.17 Coordinate differences (vectors)
=====================================
Observed coordinate differences describe relative positions of station
pairs (vectors). Contrary to the observed coordinates, the
variance-covariance matrix of the coordinate differences always
describes all three elements of the 3D vectors.
Optional attributes of empty element tag '' for describing
instrument and/or target height are
* 'from_dh = "..."' instrument height
* 'to_dh = "..."' target height
Example
=======
...
..
2.18 Example of local geodetic network
======================================
The XML input data format should be now reasonably clear from the
following sample geodetic network. This example is taken from user's
guide to Geodet/PC by Frantisek Charamza.
[image: Sketch of the example network]
XML input stream of points and observation data for the program GNU gama
3 SQL schema, SQLite and 'gama-local'
*************************************
The input data for a local geodetic network adjustment (program
'gama-local') can be strored in SQLite 3 database file. The general
information about SQLite can be found at
Input data (points, observations and other related information) are
stored in SQLite database file. Native SQLite C/C++ API is used for
reading SQLite database file. It is described at
Please note if you compile GNU Gama as described in *note Install:: and
SQLite library is not installed on your system, GNU Gama would be
compiled without SQLite support.
SQL schema ('CREATE' statements) is in 'gama-local-schema.sql' file
which is part of GNU Gama distribution and is in the 'xml' directory.
All tables for 'gama-local' are prefixed with 'gnu_gama_local_'. In the
documentation table names are referred without this prefix. For example
table 'gnu_gama_local_points' is referred as 'points'.
Database scheme used for SQLite database is also valid in other SQL
database systems. Almost every column has some constraint to ensure
correctness.
You can convert existing XML input file to SQL commands with program
'gama-local-xml2sql', for example
$ gama-local-xml2sql geodet-pc geodet-pc-123.gkf geodet-pc.sql
3.1 Working with SQLite database
================================
First of all you have to create tables for GNU Gama in SQLite database
file (here with 'db' extension, but you can choose your own, e.g.
'sqlite').
$ sqlite3 gama.db < gama-local-schema.sql
You can check created tables by following commands (fist in command
line, second in SQLite command line).
$ sqlite3 gama.db
sqlite> .tables
Output should look like this:
gnu_gama_local_clusters gnu_gama_local_descriptions
gnu_gama_local_configurations gnu_gama_local_obs
gnu_gama_local_coordinates gnu_gama_local_points
gnu_gama_local_covmat gnu_gama_local_vectors
When you have created tables you can import data. One way is to process
file with SQL statements.
$ sqlite3 gama.db < geodet-pc.sql
Another way can be filing database file in another program.
For using 'sqlite3' command you need a command line interface for SQLite
3 installed on your system (e.g. 'sqlite3' package).
3.2 Units in SQL tables
=======================
In the 'gama-local' SQLite database, distances are given in meters and
their standard deviations (rms errors) in millimeters. Angular values
are given in radians as well as their standard deviations.
Conversions between radians, gons and degrees:
rad = gon * pi / 200
rad = deg * pi / 180
gon = rad * 200 / pi
deg = rad * 180 / pi
3.3 Network SQL definition
==========================
Network definitions are stored in the 'configurations' table. This
table contains all parameters for each network such as value of a priori
reference standard deviation or orientation of the 'xy' orthogonal
coordinate system axes.
It is obvious that in one database file can be stored more networks
(configurations).
Configuration descriptions (annotation or comments) are stored
separately in table 'descriptions'. The description is split to many
records because of compatibility with various databases (not all
databases implements type 'TEXT').
Field (attribute) 'conf_id' identifies a configuration in the database.
Field 'conf_name' is used to identify configuration outside the database
(e.g. parameter in command-line when reading data from database to
'gama-local').
Table 'configurations' contains all parameters specified in tag
'' (*note Network parameters::) and also 'gama-local'
command line parameters (*note Program gama-local::). The list of all
table attributes (parameters) follows.
* 'sigma_apr' value of a priori reference standard deviation--square
root of reference variance (default value 10)
* 'conf_pr' confidence probability used in statistical tests (dafault
value 0.95)
* 'tol_abs' tolerance for identification of gross absolute terms in
project equations (default value 1000 mm)
* 'sigma_act' actual type of reference standard deviation use in
statistical tests ('aposteriori | apriori'); default value is
'aposteriori'
* 'update_cc' enables user to control if coordinates of constrained
points are updated in iterative adjustment. If test on
linerarization fails (*note Linearization::), Gama tries to improve
approximate coordinates of adjusted points and repeats the whole
adjustment. Coordinates of constrained points are implicitly not
changed during iterations. Acceptable values are 'yes', 'no',
default value is 'no'.
* 'axes_xy' orientation of axes 'x' and 'y'; value 'ne' implies that
axis 'x' is oriented north and axis 'y' is oriented east.
Acceptable values are 'ne', 'sw', 'es', 'wn' for left-handed
coordinate systems and 'en', 'nw', 'se', 'ws' for right-handed
coordinate systems (default value is 'ne').
* 'angles' 'right-handed' defines counterclockwise observed angles
and/or directions, value 'left-handed' defines clockwise observed
angles and/or directions (default value is 'left-handed').
* 'epoch' is measurement epoch. It is floating point number (default
value is '0.0').
* 'algorithm' specifies numerical method used for solution of the
adjustment. For Singular Value Decomposition set value to 'svd'.
Value 'gso' stands for block matrix algorithm GSO by Frantisek
Charamza based on Gram-Schmidt orthogonalization, value 'cholesky'
for Cholesky decomposition of semidefinite matrix of normal
equations and value 'envelope' for a Cholesky decomposition with
_envelope_ reduction of the sparse matrix. Default value is 'svd'.
* 'ang_units' Angular units of angles in 'gama-local' output. Value
'400' stands for gons and value '360' for degrees (default value is
'400'). Note that this doesn't effect units of angles in database.
For further information about angular units see *note Angular
units::.
* 'latitude' is mean latitude in network area. Default value is '50'
(gons).
* 'ellipsoid' is name of ellipsoid (*note Supported ellipsoids::).
All fields are mandatory except 'ellipsoid' field. For additional
information about handling geodetic systems in 'gama-local' see *note
Network definition::.
Example ('configuration' table contents):
conf_id|conf_name|sigma_apr|conf_pr|tol_abs|sigma_act |update_cc|...
---------------------------------------------------------------------
1 |geodet-pc|10.0 |0.95 |1000.0 |aposteriori|no |...
... axes_xy|angles |epoch|algorithm|ang_units|latitude|ellipsoid
---------------------------------------------------------------------
... ne |left-handed|0.0 |svd |400 |50.0 |
The list of 'description' table attributes follows.
* 'conf_id' is id of configuration which description (text) belongs
to.
* 'id' identifies text in a database.
* 'text' is part of configuration description. Its SQL type is
'VARCHAR(1000)'.
There can be more than one text for one configuration. All texts
related to one configuration are concatenated to one description.
Example ('description' table contents):
conf_id|indx|text
-----------------------------------------------
1 |1 |Frantisek Charamza: GEODET/PC, ...
3.4 Table 'points'
==================
* 'conf_id' is id of configuration which points belongs to.
* 'id' identifies point in a database and also in an output. It is
mandatory and it is character string (SQL type is 'VARCHAR(80)').
Point 'id' has to be unique within one configuration. In
documentation it is referred as point identification or point id.
* 'x', 'y' and 'z' coordinates of a point. Coordinate 'z' is
considered as height.
* 'txy' and 'tz' specify the type of coordinates 'x', 'y' and 'z'.
Acceptable values are 'fixed', 'adjusted' and 'constrained' (there
is no default value). For details see *note Points::.
Example (table contents):
conf_id|id |x |y |z|txy |tz
------------------------------------------
1 |201|78594.91|9498.26| |fixed |
1 |205|78907.88|7206.65| |fixed |
1 |206|76701.57|6633.27| |fixed |
1 |207| | | |adjusted|
3.5 Table 'clusters'
====================
The cluster is a group of observations with the common covariance
matrix. The covariance matrix allows to express any combination of
correlations among observations in cluster (including uncorrelated
observations, where covariance matrix is diagonal). For explanation see
*note Observation data and points::.
In the database observations are stored in three tables: 'obs',
'coordinates' and 'vectors'. Cluster's covariance matrix is stored in
table 'covmat'. Every observation, vector or coordinate in database has
to be in some cluster.
* 'conf_id' is id of configuration which cluster belongs to.
* 'ccluster' identifies a cluster within one configuration.
* 'dim' and 'band' specify dimension and bandwidth of covariance
matrix. The bandwidth of the diagonal matrix is equal to 0 and a
fully-populated covariance matrix has a bandwidth of 'dim-1'
('band' maximum possible value is 'dim-1').
* 'tag' specifies type of observations in cluster which also implies
the table where they are stored in. 'obs' and 'height-differences'
stand for 'obs' table, 'coordinates' and 'vectors' stand for
'coordinates' table and 'vectors' table respectively.
Observations, vectors and coordinates are identified by configuration id
('conf_id'), cluster id 'ccluster' and theirs index ('indx').
Observation index ('indx') has to be unique within observations of one
cluster (which belongs to one configuration). The same applies for
vectors and coordinates.
See also *note Set of observations::.
Example (table contents):
conf_id|ccluster|dim|band|tag
-----------------------------
1 |1 |3 |0 |obs
1 |4 |4 |0 |obs
3.6 Table 'covmat'
==================
Values of cluster covariance matrix are stored in 'covmat' table.
Attributes 'conf_id', 'ccluster' identifies covariance matrix. Value
position in matrix is specified by 'rind' and 'cind' fields.
* 'conf_id' is id of configuration which cluster belongs to.
* 'ccluster' is id of cluster which matrix belongs to.
* 'rind' is row number in covariance matrix
* 'cind' is column number covariance matrix
* 'val' is value itself (variance or covariance).
Values 'rind' and 'cind' have to respect 'dim' and 'band' specified in
table 'clusters'. If value in covariance matrix is not specified
(record is missing), it is considered to be zero.
Example (table contents):
conf_id|ccluster|rind|cind|val
--------------------------------
1 |1 |1 |1 |400.0
1 |1 |2 |2 |400.0
1 |1 |3 |3 |400.0
1 |4 |1 |1 |400.0
1 |4 |2 |2 |400.0
1 |4 |3 |3 |400.0
1 |4 |4 |4 |400.0
3.7 Table 'obs'
===============
Table 'obs' contains simple observations like direction or distance.
* 'conf_id' is id of configuration which cluster belongs to.
* 'ccluster' is id of cluster which observation belongs to.
* 'indx' identifies observation within cluster. It has to be
positive integer.
* 'tag' specifies a type of an observation. Allowed 'tag's follows.
* 'direction' for directions.
* 'distance' for horizontal distances.
* 'angle' for angles.
* 's-distance' for slope distances (space distances).
* 'z-angle' for zenith angles.
* 'azimuth' for azimuth angles.
* 'dh' for leveling height differences.
* 'from_id' is stand point identification. It is mandatory and it
must not differ within one cluster for observations with 'tag =
'direction'' .
* 'to_id' is target identification (mandatory).
* 'to_id2' is second target identification. It is valid and
mandatory only for angles ('tag = 'angle'').
* 'val' is observation value. It is mandatory for all observation
types.
* 'stdev' is value of standard deviation. It is used when variance
in covariance matrix is not specified.
* 'from_dh' is value of instrument height (optional).
* 'to_dh' is value of reflector/target height (optional).
* 'to_dh2' is value of second reflector/target height (optional). It
is valid only for angles.
* 'dist' is distance of leveling section. It is valid only for
height-differences ('tag = 'dh'').
* 'rejected' specifies whether observation is rejected (passive) or
not. Value '0' stand for not rejected, value '1' for rejected. It
is mandatory. Default value is '0'.
Example (table contents without empty columns):
conf_id|ccluster|indx|tag |from_id|to_id|val |rejected
---------------------------------------------------------------------
1 |1 |1 |direction|201 |202 |0.0 |0
1 |1 |2 |direction|201 |207 |0.817750284544|0
1 |1 |3 |direction|201 |205 |2.020073921388|0
3.8 Table 'coordinates'
=======================
Table 'coordinates' contains control (known) coordinates.
* 'conf_id' is id of configuration which cluster belongs to.
* 'ccluster' is id of cluster which coordinates belongs to.
* 'indx' identifies coordinates within cluster. It has to be
positive integer.
* 'id' is point identification.
* 'x', 'y' and 'z' are coordinates.
* 'rejected' specifies whether observation is rejected (passive) or
not. Value '0' stand for not rejected, value '1' for rejected.
Default value is '0'.
See also *note Control coordinates::.
3.9 Table 'vectors'
===================
Table 'vectors' contains coordinate differences (vectors).
* 'conf_id' is id of configuration which cluster belongs to.
* 'ccluster' is id of cluster which vector belongs to.
* 'indx' identifies vector within cluster. It has to be positive
integer.
* 'from_id' is point identification. It identifies initial point.
* 'to_id' is point identification. It identifies terminal point.
* 'dx', 'dy' and 'dz' are coordinate differences.
* 'from_dh' is value of initial point height. It is optional.
* 'to_dh' is value of terminal point height. It is optional.
* 'rejected' integer default 0 not null,
See also *note Coordinate differences::.
3.10 Example of local geodetic network in SQL
=============================================
Providing complete example would be reasonable because of its extent.
However, you can obtain example by following these instructions:
Create a file with XML representation of network by copy and paste
example from *note Example:: to a new file. Note that file should start
with '' (no whitespace). Alternatively you can
use existing XML file from collection of sample networks (see *note
Download::). Then you can convert your XML file (here
'example_network.xml') to SQL statements by program 'gama-local-xml2sql'
(the path depends on your Gama installation).
$ gama-local-xml2sql example_net example_network.xml example_network.sql
Now you have example network (configuration 'example_net') in the form
of SQL 'INSERT' statements in the file 'example_network.sql'.
Another representations you can create and fill SQLite database (for
details see *note Working with SQLite database::):
$ sqlite3 examples.db < gama-local-schema.sql
$ sqlite3 examples.db < example_network.sql
$ sqlite3 examples.db
Once you have SQLite database, you can work with it from SQLite command
line. You can get nice output by executing following commands.
sqlite> .mode column
sqlite> .nullvalue NULL
sqlite> SELECT * FROM gnu_gama_local_configurations;
sqlite> SELECT * FROM gnu_gama_local_points;
sqlite> SELECT * FROM gnu_gama_local_clusters;
sqlite> SELECT * FROM gnu_gama_local_covmat;
sqlite> SELECT * FROM gnu_gama_local_obs;
Or you can get database dump ('CREATE' and 'INSERT' statements) by
sqlite> .dump
If it is not enough for you, you can try one of GUI tools for SQLite.
4 Network adjustment with 'gama-local'
**************************************
Adjustment of local geodetic network is a classical case of _adjustment
of indirect observations._ After estimation of approximate values of
unknown parameters (coordinates of points) and linearization of
functions describing relations between observations and parameters we
solve linear system of equations
(1) Ax = b + v,
where 'A' is coefficient matrix, 'b' is vector of absolute terms (right
hand side) and 'v' is vector of residuals. This system is (generally)
overdetermined and we seek the solution 'x' satisfying the basic
criterion of Least Squares
(2) v'Pv = min,
where 'P' is weight matrix. This criterion unambiguously defines the
shape of adjusted network.
In the case of _free network_ the system (1) is singular (matrix 'A' has
linearly dependent columns) and we have to define second regularization
criterion
(3) \sum x_i^2 = min, for all selected i
stating that at the same time we demand that the sum of squares
corrections of selected parameters is minimal (corrections of unknown
parameters with indexes from the set of all selected unknowns.
Geometrically this criterion is equivalent to adjustment of the network
according to (2) with simultaneous transformation to the selected set of
fiducial points. This transformation does not change the shape of
adjusted network.
Often it is advantageous to work with a _homogenized system,_ ie. with
the system of project equations in which coefficient of each row and
absolute term are multiplied by square root of the weight of
corresponding observation.
(4) ~A x = ~b,
where ~A = P^{1/2} A, ~b = P^{1/2} A. Symbol P^{1/2} denotes diagonal
matrix of square roots of observation weights (or Cholesky decomposition
of covariance matrix in the case of correlated observations). To
criterion (2) corresponds in the case of homogenized system criterion
(5) ~v'v = min.
Normal equations are clearly equivalent for both systems.
4.1 Approximate coordinates
===========================
For computation of coefficients in system (1) (ie. during linearization)
we need, first of all, an estimate of approximate coordinates of points
and approximate values of orientations of observed directions sets.
Approximate values of unknown parameters are usually not known and we
have to compute them from the available observations. For approximate
value of orientation program 'gama-local' uses median of all estimates
from the given set of directions to the points with known coordinates.
Median is less sensitive to outliers than arithmetic mean which is
normally used for approximate estimate of orientations
During the phase of computation of approximate coordinate of points,
program 'gama-local' walks through the list of computed points and for
each point gathers all determining elements pointing to points with
known or previously computed coordinates. Determining elements are
*outer bearing* (oriented half-line) starting from the point with
known coordinates and pointing to the computed point
*distance* between given and computed points
*inner angle* with vertex in the computed point and arms
intersecting given points
For all combinations of determining elements program 'gama-local'
computes intersections and estimates approximate coordinates as the
median of all available solutions.
If at least one point was resolved while iterating through the list, the
whole cycle is repeated.
If no more coordinates can be solved using intersections and points with
unknown coordinates are remaining, program tries to compute coordinates
of unresolved points in a local coordinates system and obtain their
coordinates using similarity transformation. If a transformation
succeeds to resolve coordinates at least one computed point and there
are still some points without coordinates left, the whole process is
repeated. Classes for computation of approximate coordinates have been
written by Jiri Vesely.
If program 'gama-local' fails to compute approximate coordinates of some
of the network points, they are eliminated from the adjustment and they
are listed in the output listing.
With the outlined strategy, program 'gama-local' is able to estimate
approximate coordinates in most of the cases we normally meet in
surveying profession. Still there are cases in which the solution
fails. One example is an inserted horizontal traverse with sets of
observed direction on both ends but without a connecting observed
distance. The solution of approximate coordinates can fail when there
is a number of gross error for example resulting from confusion of point
identifications but in normal situations, leaving computation of
approximate coordinates on program 'gama-local' is recommended.
Example
=======
Computation of approximate coordinates of points
************************************************
Number of points with given coordinates: 2
Number of solved points : 2
Number of observations : 4
-----------------------------------------------------
Successfully solved points : 0
Remaining unsolved points : 2
List of unresolved points
*************************
422
424
4.2 Gross absolute terms
========================
One of parameters in XML input of program 'gama-local' is tolerance
'tol-abs' for detecting of gross absolute terms in project equations.
Observations with outlying absolute terms are always excluded from
adjustment.
For measured distances program tests difference between observed value
d_i and distance computed from approximate coordinates d_0
|d_i - d_0| > tol-abs,
for observed directions program 'gama-local' tests transverse deviation
corresponding to absolute term b_i from project equations (1)
| b_i | d_0 > tol-abs
and similarly for angles, program tests the greater of two deviations
corresponding to left and right distances (left and right arm of the
angle)
|b_i| max{ d_{0_l}, d_{0_r} } > tol-abs.
Default value of parameter 'tol-abs' is 1000 mm.
Example
=======
Outlying absolute terms in project equations
********************************************
i standpoint target observed absolute
=========================================== value ===== term ==
2 103 104 dir. 301.087900 -9989.1
Observations with outlying absolute terms removed
4.3 Parameters of statistical analysis
======================================
Program 'gama-local' uses two basic statistical parameters
* confidence probability P (default value is 95%, see input XML
parameter 'conf-pr') and
* actual type of reference standard deviation m0_a (parameter
'sigma-act').
Confidence probability determines significance level on which
statistical tests of adjusted quantities are carried. Actual type of
reference standard deviation m0_a specifies whether during statistical
analysis we use an a priori reference standard deviation m0 or an a
posteriori estimate m0'. On the type of actual reference standard
deviation depends the choice of density functions of stochastic
quantities in statistical analysis of the adjustment.
*A priori reference standard deviation m0* is an estimate of the
standard deviation of an observation with the unit weight.
Numerically it is a scaling factor used in calculation of the
weights. If we change m0, only the sum of weighted residuals
squares is changed and all adjustment results remain the same
(there is just one least squares solution). m0 can be selected in
cases when we know its value in advance and with sufficient
reliability. Another situation when m0 is used are networks with
low number of degrees of freedom (poorly overdetermined systems) or
when veen degrees of freedom is zero. Examples may be analysis of
network models etc.
*A posteri estimate of reference standard deviation m0'* is used in
cases when a priori value of reference standard deviation m0 is not
known and when degrees of freedom is sufficiently high and reliable
for empirical estimate of m0'.
The standard deviantion of an adjusted quantity is computed in
dependence of the choice of actual type of reference standard deviation
m0_a according to formula
m0_a sqrt(q)
where q is the weight coefficient of the corresponding adjusted unknown
parameter or observation. Apart from the standard deviation, program
'gama-local' computes for the adjusted quantity its _confidence
interval_ in which its real value is located with the probability P.
4.4 Test on the reference standard deviation
============================================
Null hypothesis H_0: m0 = m0' is tested versus alternative hypothesis
H_1: m0 \neq m0'. Test criterion is ratio of a posteriori estimate of
reference standard deviation
m0' = sqrt( v'P v / r).
and a priori reference standard deviation m0 (input data parameter
'm0-apr'). For given significance level \alpha lower and upper bounds
of interval (L, U) are computed so, that if hypothesis H_0 is true,
probabilities P(m0'/m0 \le D) and P(m0'/m0 \ge H) are equal to \alpha/2.
Lower and upper bounds of the interval are computed as
L = sqrt((Chi^2_{1-alpha/2,r})/r),
U = sqrt((Chi^2_{ alpha/2 ,r})/r).
Probability
P(L < m0'/m0 < U) = conf-pr
is by default 95%, this corresponds to 5% confidence level test.
Exceeding the upper limit H of the confidence interval can be caused
even by a single gross error (one outlying observation). Method of
Least Squares is generally very sensitive to presence of outliers.
Safely can be detected only one observation whose elimination leads to
maximal decrease of a posteriori estimate of reference standard
deviation
(6) m0'' = sqrt{(v'P v - delta)/(r-1)},
delta = max(v_i^2/q_vi),
where
(7) q_vi = 1/p_i - q_Li
is weight coefficient of i-th residual. If the set of observations
contains only one gross error, the outlying observation is likely to be
detected, but this can not be guaranteed.
In addition, program 'gama-local' computes a posteriori estimate of
reference standard deviation separately for horizontal distances and
directions and/or angles after formula from
m0'_t = sqrt(sum{~v^2_it}) / sum{~q_vi}), t=d,s,
where symbol t denotes observed distances, directions and/or angles.
Example
=======
m0 apriori : 10.00
m0' empirical: 9.64 [pvv] : 3.43560e+03
During statistical analysis we work
- with empirical standard deviation 9.64
- with confidence level 95 %
Ratio m0' empirical / m0 apriori: 0.964
95 % interval (0.773, 1.227) contains value m0'/m0
m0'/m0 (distances): 0.997 m0'/m0 (directions): 0.943
Maximal decrease of m0''/m0 on elimination of one observation: 0.892
Maximal studentized residual 2.48 exceeds critical value 1.95
on significance level 5 % for observation #35
4.5 Information on points
=========================
Program 'gama-local' lists separately review of coordinates of fixed and
adjusted points; adjusted _constrained_ coordinates are marked with '*';
see equation (3). Adjusted coordinate standard deviations m_x and m_y,
and values for computing confidence intervals are given in the listing
of adjusted coordinates (*note Statistical analysis::). In the review
index i is the index of unknown x_i from the system of project equations
(1) corresponding to the point coordinates x and y.
Example
=======
Fixed points
************
point x y
========================================
1 1054980.484 644498.590
2 1054933.801 643654.101
Adjusted coordinates
********************
i point approximate correction adjusted std.dev conf.i.
====================== value ====== [m] ====== value ========== [mm] ===
422
2 x 1055167.22747 -0.00510 1055167.22237 2.7 5.4
3 y 644041.46119 0.00023 644041.46142 2.5 5.1
424
4 X * 1055205.41198 -0.00056 1055205.41142 3.1 6.3
5 Y * 644318.24425 -0.00125 644318.24300 3.6 7.2
For adjusted points, program summarizes information on standard
ellipses, confidence ellipses, mean square positional errors (m_p), mean
coordinate errors (m_xy) and coefficients g characterizing position of
approximate coordinates with regard to the confidence ellipse.
Example
=======
Mean errors and parameters of error ellipses
********************************************
point mp mxy mean error ellipse conf.err. ellipse g
========== [mm] == [mm] ==== a [mm] b alpha[g] ==== a' [mm] b' ========
422 3.6 2.6 2.7 2.5 187.0 6.8 6.4 0.8
424 4.7 3.4 3.7 2.9 131.8 9.5 7.4 0.2
403 5.7 4.0 4.3 3.6 78.9 11.0 9.3 1.1
Mean square positional error m_p and mean coordinate error (m_xy) are
computed as
m_p = sqrt(m_y^2 + m_x^2), m_xy = m_p / sqrt(2),
where m_y^2 and m_x^2 are squares of standard deviations (variances) of
adjusted points coordinates.
Semimajor and semiminor axes of standard ellipse are denoted as a and b
in the listing, bearing of semimajor axis is denoted as \alpha and they
are computed from covariances of adjusted coordinates
a = sqrt(1/2(cov_yy + cov_xx + c),
b = sqrt(1/2(cov_yy + cov_xx - c),
c = sqrt( (cov_xx - cov_yy)^2 + 4(cov_xy)^2 ),
tan 2alpha = 2(cov_xy) / (cov_xx - cov_yy).
The angle \alpha (the bearing of semimajor axis) is measured clockwise
from X axis.
Probability that standard ellipse covers real position of a point is
relatively low. For this reason program 'gama-local' computes extra
_confidence ellipse_ for which the probability of covering real point
position is equal to the given confidence probability. Both ellipsy are
located in the same center, they share the same bearing of semimajor
axes and they are similar. For lengths of their semi-axis holds
a' = k_p a, b' = k_p b,
where k_p is a coefficient computed for the given probability P as
defined in *note Statistical analysis::.
[image: Approximate position of adjusted point with
regard to confidence ellipse]
Position of approximate coordinates of an adjusted point with respect to
its confidence ellipse are expressed by a coeeficient g Three cases are
possible
g < 1 approximate coordinates of adjusted point are located inside
the confidence ellipse
g = 1 approximate coordinates of adjusted point are located on the
confidence ellipse
g > 1 approximate coordinates of adjusted point are outside the
confidence ellipse
The coefficient g is calculated from formula
g = sqrt( (a_0 / a')^2 + (b_0/b')^2 )
where
b_0 = delta_y cos(alpha) - delta_x sin(alpha),
a_0 = delta_y sin(alpha) - delta_x cos(alpha)
symbol \delta is used for correction of approximate coordinates and
\alpha is bearing of confidence ellipse semimajor axis.
If network contains sets of observed directions, program writes
information on corresponding adjusted orientations, standard deviations
and confidence intervals. Index i is the same as in the case of
adjusted coordinates the index of i-th adjusted unknown in the project
equations.
Example
=======
Adjusted bearings
*****************
i standpoint approximate correction adjusted std.dev conf.i.
==================== value [g] ==== [g] === value [g] ======= [cc] ===
1 1 296.484371 -0.000917 296.483454 5.1 10.3
10 2 96.484371 0.000708 96.485079 5.1 10.4
21 403 20.850571 -0.001953 20.848618 8.8 17.7
4.6 Adjusted observations and residuals
=======================================
In the review of adjusted observations program 'gama-local' prints index
of the observation, index of the row in matrix 'A' in the system (1),
identifications of standpoint and target point, type of the observation,
its approximate and adjusted value, standard deviation and confidence
interval.
Example
=======
Adjusted observations
*********************
i standpoint target observed adjusted std.dev conf.i.
===================================== value ==== [m|g] ====== [mm|cc] ==
1 1 2 dis. 845.77700 845.77907 3.0 6.1
2 422 dir. 28.205700 28.205613 5.1 10.3
3 424 dir. 60.490600 60.491359 6.7 13.6
Review of residuals serves for analysis of observations and containts
values of normalized or studentized residuals (depending on type of m0_a
used) and three characteristics. Theese are coefficient 'f' identifying
weak network elements and estimates of real error of observation 'e-obs'
and real error of its adjusted value 'e-adj', see definition in the
following text.
If normalized or studentized residual exceeds critical value for the
given confidence probability, it is marked in the review with symbol 'c'
(critical) and maximal normalized or studentized residual is marked with
symbol 'm'.
Example
=======
Residuals and analysis of observations
**************************************
i standpoint target f[%] v |v'| e-obs. e-adj.
======================================== [mm|cc] =========== [mm|cc] ===
1 1 2 dir. 47.4 9.170 1.1 12.7 3.5
2 422 dir. 47.0 -0.873 0.1 -1.2 -0.3
3 424 dir. 30.3 7.588 1.1 14.8 7.2
4.7 Identification of weak network elements
===========================================
When planning observations in a geodetic network we always try to
guarantee that all observed elements are checked by other measurements.
Only with redundant measurements it is possible to adjust observations
and possibly remove blunders that might otherwise totaly corrupt the
whole set of measurements. Apart from sufficient number of redundant
observations the degree of control of single observed elements is given
by the network configuration, ie. its geometry.
Less controlled observations represent weak network elements and they
can in extreme cases even disable detection of gross observational
errors as it is in the case of uncontrolled observations. There are two
limit cases of observation control
*fully controlled observation* as is for example an observed
distance between two fixed points (standard deviation of the
adjusted element is zero; standard deviation of the residual equals
to the standard deviation if the observation) and
*uncontrolled observations* as is a free polar bar for example
(standard deviation of adjusted value is equal to standard
deviation of observed quantity; residual and standard deviation of
the residual are zero).
Weakly controlled or uncontrolled observations can result even from
elimination of certain suspisios observations during analysis of
adjusment.
Standard deviation of adjusted observations is less than standard
deviation of the measurement. Degree of observation control in network
is defined as coefficient
(8) f = 100 (m_l - m_L)/m_l,
where m_l is standard deviation of observed quantity and m_L is standard
deviation computed from a posteriori reference standard deviation m0.
We consider observed network element to be
*uncontrolled* if f < 0.1 (in listing marked with letter 'u'),
*weakly controlled* if 0.1 \le f < 5 (in listing marked with letter
'w').
4.8 Estimation of real errors
=============================
Acording to previous section we can consider an observation to be
controlled if its coefficient f > 0.1. Any controlled observation can
be eliminated from the network without corrupting the network
consistency--network reduced by one controlled observation can be
adjusted and all unknown parameters can be compute without the
eliminated observation.
Estimate of real error of i-th observation is defined as
(9) e_li = L^red_i - l_i,
where e_li is value of i-th observation and is value of i-th network
element computed from adjusted coordinates and/or orientations of the
reduced network. Similarly is defined the estimate of real error of a
residual
(10) e_vi = L^red_i - L_l.
Adjustment results are the best statistical estimate of unknown
parameters that we have. This holds true even for adjustment of
_reduced_ network which is not influenced by real error of i-th
observation. On favourable occasions differences (9) and (10) can help
to detect blunders but to interpret these estimates as _real errors_ is
possible only with substantial exaggeration. These estimates fail when
there are more than one significant observational error. Generally
holds tha the weaker the element is controlled in netowrk the less
reliable these estimates are.
Estimate of real error of an observation computes program 'gama-local'
as
e_li = v_i/(p_i q_vi)
and estimate of real error of a residual as
e_vi = e_li - v_i.
4.9 Test on linearization
=========================
Mathematical model of geodetic network adjustment in 'gama-local' is
defined as a set of known real-valued differentiable functions
(11) L^* = f(X^*)
where L^* is a vector of theoretical correct observations and X^* is a
vector of correct values of parameters. For the given sample set of
observations 'L' and the unknown vector of residuals 'v' we can express
the estimate of parameters 'X' as a nonlinear set of equations
(12) L + v = f(X).
With approximate values 'X_0' of unknown parameters
(12) X = X_0 + x
we can linearize the equations (12)
L + v = f(X_0) + f'(X_0)x.
yielding the linear set of equations (1)
Unknown parameters in 'gama-local' mathematical model are points
coordinates and orientation angles (transforming observed directions to
bearings). The observables described by functions (12) belong into two
classes
*linear observables*: horizontal and slope distances, height
differences, control coordinates and vectors (coordinate
differences),
*angular observables*: directions, horizontal and zenith angles.
Internally in 'gama-local' unknown corrections to linear observables are
computed in millimeters and corrections to angular observables in
centigrade seconds. To reflect the internal units in used all partial
derivatives of angular observables by coordinates are scaled by factor
2000/pi.
When computing coefficients of project equations (1) we expect that
approximate coordinates of points are known with sufficient accuracy
needed for linearization of generally nonlinear relations between
observations and unknown paramters. Most often this is true but not
always and generally we have to check how close our approximation is to
adjusted parameters.
Generally we check linearization in adjustment by double calculation of
residuals
v^I = Ax - b,
v^II = ~l(~x) - l,
Program 'gama-local' similarly computes and tests differences in values
of adjusted observations once computed from residuals and once from
adjusted coordinates. For measured directions and angles 'gama-local'
computes in addition transverse deviation corresponding to computed
angle difference in the distance of target point (or the farther of two
targets for angle). As a criterion of bad linearization is supposed
positional deviation greater or equal to 0.0005 millimetres.
Example
=======
Test of linearization error
***************************
Diffs in adj. obs from residuals and from adjusted coordinates
**************************************************************
i standpoint target observed r difference
================================= value = [mm|cc] = [cc] == [mm]=
2 3022184030 3022724008 dist. 28.39200 -7.070 -0.003
3 3022724002 dist. 72.30700 -18.815 -0.001
7 3000001063 dir. 286.305200 11.272 -0.002 -0.001
8 3022724008 dir. 357.800600 -23.947 0.037 0.002
From the practical point of view it might seem that the tolerance 0.0005
mm for detecting poor linearization is too strict. Its exceeding in
program 'gama-local' results in repeated adjustment with substitute
adjusted coordinates for approximate. Given tolerance was chosen so
strict to guarantee that listed output results would never be influenced
by linearization and could serve for verification and testing of
numerical solutions produced by other programs.
Implicitly coordinates of constrained points are not changed in
iterative adjustments. This feature can be changed in XML input data by
setting '' (*note
Network parameters::).
Iterated adjustement with successive improvement of approximate
coordinates converges usually even for gross errors in initial estimates
of unknown coordinates. If the influence of linearization is detected
after adjustment, quite often only one iteration is sufficient for
recovering.
For any automatically controlled iteration we have to set up certain
stopping criterion independent on the convergence and results obtained.
Program 'gama-local' computes iterated adjustment three times at
maximum. If the bad linearization is detected even after three
readjustments it signals that given network configuration is somehow
suspicious.
5 Data structures and algorithms
********************************
5.1 Observation data and points
===============================
The Gama observation data structures are designed to enable adjustment
of any combination of possibly correlated observations. At its very
early stage Gama was limited to adjustment of uncorrelated observations.
Only directions and distances were available and observable's weight was
stored together with the observed value in a single object. A single
array of pointers to observation objects was sufficient for handling all
observations. So called _orientation shifts_ corresponding to
directions measured form a point were stored together with coordinations
in _point objects_.
[image: Schema of observation data structures]
To enable adjustment of possibly correlated observations (like angles
derived from observed directions or already adjusted coordinates from a
previous adjustment) Gama has come with the concept of _clusters_.
Cluster is an object with a common variance-covariance matrix and a list
of pointers to observation objects (distances, directions, angles,
etc.). Weights were removed from observation objects and replaced with
a pointer to the cluster to which the observation belong. All clusters
are joined in a common object 'ObservationData'; similarly to
observations, each cluster contains a pointer to its parent 'Observation
Data' object. _Orientation shifts_ were separated from coordinates and
are stored in the cluster containing the bunch of directions and thus
number of orientations is not limited to one for a point.
This organisation of observational information has proved to be
effective. Template classes 'ObservationData' and 'Cluster' are used as
base classes both in 'gama-local' and 'gama-g3'
template
class ObservationData
{
public:
ClusterList CL;
ObservationData();
ObservationData(const ObservationData& cod);
~ObservationData();
ObservationData& operator=(const ObservationData& cod);
template void for_each(const P& p) const;
};
template
class Cluster
{
public:
const ObservationData* observation_data;
ObservationList observation_list;
typename Observation::CovarianceMatrix covariance_matrix;
Cluster(const ObservationData* od);
virtual ~Cluster();
virtual Cluster* clone(const ObservationData*) const = 0;
double stdDev(int i) const;
int size() const;
void update();
int activeCount() const;
typename Observation::CovarianceMatrix activeCov() const;
};
The following template class 'PointBase' for handling point information
is used in 'gama-g3'. The template class 'PointBase' relies internally
on 'std::map' container but comes with its own interface (in
'gama-local' 'std::map' was used directly for storing points).
template
class PointBase
{
typedef std::map Points;
public:
PointBase();
PointBase(const PointBase& cod);
~PointBase();
PointBase& operator=(const PointBase& cod);
void put(const Point&);
void put(Point*);
Point* find(const typename Point::Name&);
const Point* find(const typename Point::Name&) const;
void erase(const typename Point::Name&);
void erase();
class const_iterator;
const_iterator begin();
const_iterator end ();
class iterator;
iterator begin();
iterator end ();
};
Template classes 'ObservationData' and 'PointBase' are defined in
namespace 'GNU_gama' and are located in the source directory 'gnu_gama'.
5.2 Supported ellipsoids
========================
id a b, 1/f, f description
airy 6377563.396 6356256.910 Airy ellipsoid 1830 [4]
airy_mod 6377340.189 6356034.446 Modified Airy [4]
apl1965 6378137 298.25 Appl. Physics. [4]
1965
andrae1876 6377104.43 300.0 Andrae 1876 [4]
(Denmark, Iceland)
australian 6378160 298.25 Australian National [3]
1965
bessel 6377397.15508 6356078.96290 Bessel ellipsoid [1]
1841
bessel_nam 6377483.865 299.1528128 Bessel 1841 [4]
(Namibia)
clarke1858a 6378361 6356685 Clarke ellipsoid [3]
1858 1st
clarke1858b 6378558 6355810 Clarke ellipsoid [3]
1858 2nd
clarke1866 6378206.4 6356583.8 Clarke ellipsoid [3]
1866
clarke1880 6378316 6356582 Clarke ellipsoid [3]
1880
clarke1880m 6378249.145 293.4663 Clarke ellipsoid [4]
1880 (modified)
cpm1799 6375738.7 334.29 Comm. des Poids et [4]
Mesures 1799
delambre 6376428 311.5 Delambre 1810 [4]
(Belgium)
engelis 6378136.05 298.2566 Engelis 1985 [4]
everest1830 6377276.345 300.8017 Everest 1830 [4]
everest1848 6377304.063 300.8017 Everest 1948 [4]
everest1856 6377301.243 300.8017 Everest 1956 [4]
everest1869 6377295.664 300.8017 Everest 1969 [4]
everest_ss 6377298.556 300.8017 Everest (Sabah and [4]
Sarawak)
fisher1960 6378166 298.3 Fisher 1960 (Mercury [3]
Datum) [4]
fisher1960m 6378155 298.3 Modified Fisher 1960 [3]
[4]
fischer1968 6378150 298.3 Fischer 1968 [4]
grs67 6378160 298.2471674270 GRS 67 (IUGG 1967) [4]
grs80 6378137 298.257222101 Geodetic Reference [1]
System 1980
hayford 6378388 297 Hayford 1909 [1]
(International) [3]
helmert 6378200 298.3 Helmert ellipsoid [3]
1906
hough 6378270 297 Hough [4]
iau76 6378140 298.257 IAU 1976 [4]
international 6378388 297 International 1924 [1]
(Hayford 1909) [3]
kaula 6378163 298.24 Kaula 1961 [4]
krassovski 6378245 298.3 Krassovski ellipsoid [1]
1940
lerch 6378139 298.257 Lerch 1979 [4]
mprts 6397300 191.0 Maupertius 1738 [4]
mercury 6378166 298.3 Mercury spheroid [3]
1960
merit 6378137 298.257 MERIT 1983 [4]
new_intl 6378157.5 6356772.2 New International [4]
1967
nwl1965 6378145 298.25 Naval Weapons Lab., [4]
1965
plessis 6376523 6355863 Plessis 1817 [4]
(France)
se_asia 6378155 6356773.3205 Southeast Asia [4]
sgs85 6378136 298.257 Soviet Geodetic [4]
System 85
schott 6378157 304.5 Schott 1900 spheroid [3]
sa1969 6378160 298.25 South American [3]
Spheroid 1969
walbeck 6376896 6355834.8467 Walbeck [4]
wgs60 6378165 298.3 WGS 60 [4]
wgs66 6378145 298.25 WGS 66 [4]
wgs72 6378135 298.26 WGS 72 [4]
wgs84 6378137 298.257223563 World Geodetic [1]
System 1984
[1] Milos Cimbalnik - Leos Mervart: Vyssi geodezie 1, 1997,
Vydavatelstvi CVUT, Praha
[2] Milos Cimbalnik: Derived Geometrical Constants of the Geodetic
Reference System 1980, Studia geoph. et geod. 35 (1991), pp.
133-144, NCSAV, Praha
[3] Glossary of the Mapping Sciences, Prepared by a Joint Committe of
the American Society of Civil Engineers, American Congress on
Surveying and Mapping and American Society for Photogrammetry and
Remote Sensing (1994), USA, ISBN 1-57083-011-8, ISBN
0-7844-0050-4
[4] Gerald Evenden: proj - forward cartographic projection filter
(rel. 4.3.3), http://www.remotesensing.org/proj
5.3 Transformation from spatial to geographical coordinates
===========================================================
Spatial coordinates (X, Y, Z) can be easily computed from geographical
ellipsoidal coordinates (B, L, H), where B is geographical latitude, L
geographical longitude and H is elliposidal height, as
X = (N + H) cos B cos L
Y = (N + H) cos B sin L
Z = (N(1-e^2) + H)sin B
where N = a/sqrt(1 - e^2 sin^2 B) is the radius of curvature in the
prime vertical, e^2 = (a^2 - b^2)/a^2 is the first eccentricity for the
given rotational ellipsoid (spheroid) with semi-major axis a and
semi-minor axis b.
In the case of coordiante transformation from (X, Y, Z) to (B, L, H),
the longitude is given by the formula
tan L = Y / X.
Now we can introduce
D = sqrt(X^2 + Y^2),
so that the cartesian system become (D, Z). Coordinates B and H are
then usually computed by iteration with some starting value of B_0, for
example
tan B_0 = Z/D/(1 - e^2),
tan B = Z/D + N/(N+H) e^2 tan B, H = D / cos B = Z / sin B - N(1-e^2)
B. R. Bowring described a closed formula(1) that is more effective and
sufficiantly accurate and that is used in GNU Gama.
[image: Spatial and geographical coordinates]
The centre of curvature C of the spheroid corresponding to P' is the
point
(e^2 a cos^3 u, -e'^2 b sin^3 u)),
where e'^2 = (a^2 - b^2)/b^2 is second eccentricity and u is the
parametric latitude of the point P', (1-e^2)N sin B = b sin u. Therefore
tan B = (Z + e'^2 b sin^3 u) / (D - e^2 a cos^3 u).
This is clearly an iterative solution; but it has been found that this
formula is extremely accurate using the single first approximation for u
for the tan u = (Z/D)(a/b). Maximum error in earth bound region is 3e-8
of sexadecimal arc seconds (5e-7 millimetres); maximum is 0.0018" (0.1
millimetres) at height H = 2a.
---------- Footnotes ----------
(1) B. R. Bowring: Transformation from spatial to geographical
coordinates, Survey Review XXIII, 181, July 1976
5.4 Class 'g3::Model'
=====================
g3::model documentation shall come here ...
namespace GNU_gama { namespace g3 {
class Model {
public:
typedef GNU_gama::PointBase PointBase;
typedef GNU_gama::ObservationData ObservationData;
PointBase *points;
ObservationData *obs;
GNU_gama::Ellipsoid ellipsoid;
Model();
~Model();
Point* get_point(const Point::Name&);
void write_xml(std::ostream& out) const;
void pre_linearization();
}}
Appendix A Copying This Manual
******************************
A.1 GNU Free Documentation License
==================================
Version 1.1, March 2000
Copyright (C) 2000 Free Software Foundation, Inc.
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
0. PREAMBLE
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modifications made by others.
This License is a kind of "copyleft", which means that derivative
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We have designed this License in order to use it for manuals for
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A.1.1 ADDENDUM: How to use this License for your documents
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Copyright (C) YEAR YOUR NAME.
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their use in free software.
Concept Index
*************
* Menu:
* '': Angles. (line 819)
* '': Azimuths. (line 909)
* '': Control coordinates.
(line 991)
* '': Network description.
(line 491)
* '': Directions. (line 758)
* '': Horizontal distances.
(line 788)
* '': Network definition. (line 435)
* '': Height differences. (line 947)
* '': Network definition. (line 435)
* '': Set of observations.
(line 667)
* '': Network parameters. (line 506)
* '': Points. (line 608)
* '': Points and observations.
(line 561)
* '': Slope distances. (line 851)
* '': Coordinate differences.
(line 1018)
* '': Zenith angles. (line 880)
* absolute terms: Gross absolute terms.
(line 1752)
* analysis, statistical: Statistical analysis.
(line 1792)
* angle: Angles. (line 819)
* angle, zenith: Zenith angles. (line 880)
* azumuth: Azimuths. (line 909)
* contributors: Contributors. (line 345)
* coordinate differences: Coordinate differences.
(line 1018)
* coordinates, observed: Control coordinates.
(line 991)
* description, network: Network description.
(line 491)
* deviation, reference standard: Standard deviation. (line 1839)
* deviation, standard: Standard deviation. (line 1839)
* difference, height: Height differences. (line 947)
* direction: Directions. (line 758)
* distance, horizontal: Horizontal distances.
(line 788)
* distance, slope: Slope distances. (line 851)
* download: Download. (line 137)
* FDL, GNU Free Documentation License: GNU Free Documentation License.
(line 2583)
* 'gama-local': Program gama-local. (line 212)
* gross absolute terms: Gross absolute terms.
(line 1752)
* height differences: Height differences. (line 947)
* height, difference: Height differences. (line 947)
* horizontal distance: Horizontal distances.
(line 788)
* horizontal, distance: Horizontal distances.
(line 788)
* information on points: Information on points.
(line 1911)
* install: Install. (line 155)
* network description: Network description.
(line 491)
* network parameters: Network parameters. (line 506)
* observations, Points: Points and observations.
(line 561)
* observations, set: Set of observations.
(line 667)
* observed coordinates: Control coordinates.
(line 991)
* observed, coordinates: Control coordinates.
(line 991)
* parameters of statistical analysis: Statistical analysis.
(line 1792)
* parameters, network: Network parameters. (line 506)
* point: Points. (line 608)
* points: Information on points.
(line 1911)
* points and observations: Points and observations.
(line 561)
* points, observations: Points and observations.
(line 561)
* prologue: Prologue. (line 420)
* reductions, horizontal and zenith angles: Reductions of horizontal and zenith angles.
(line 318)
* reference standard deviation: Standard deviation. (line 1839)
* Reporting bugs: Reporting bugs. (line 332)
* set of observations: Set of observations.
(line 667)
* set, observations: Set of observations.
(line 667)
* slope distance: Slope distances. (line 851)
* slope, distance: Slope distances. (line 851)
* standard deviation: Standard deviation. (line 1839)
* statistical analysis: Statistical analysis.
(line 1792)
* terms, absolute: Gross absolute terms.
(line 1752)
* test on the reference standard deviation: Standard deviation.
(line 1839)
* vector: Coordinate differences.
(line 1018)
* Windows, precompiled binaries: Precompiled binaries for Windows.
(line 205)
* zenith angle: Zenith angles. (line 880)
* zenith, angle: Zenith angles. (line 880)