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# GNU Gama 1.14

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# 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.

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.

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GNU Gama can be found in the subdirectory /gnu/gama/ on your favourite GNU mirror or checked-out from the GIT. See our project page at savannah 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 

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## 1.2 Install

GNU Gama is developed and tested under Debian GNU/Linux (http://www.debian.org/). 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 (see section 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.

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## 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 (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: 1.14 / 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 --encoding utf-8 | iso-8859-2 | iso-8859-2-flat | cp-1250 | cp-1251 --angles 400 | 360 --latitude --ellipsoid --text adjustment_results.txt --xml adjustment_results.xml --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 --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.  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ### 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.  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 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.  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 SQL schema, SQLite and gama-local.  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] # 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 <xxx> and </xxx> and empty-element tags <xxx/>. The syntax of XML input format is defined in the Document Type Definition (DTD) at and can formally be validated independently on the program gama-local. 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 (see section Angular units). At the end of this chapter an example of the gama-local XML input data object is given.  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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.  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.2 Prologue XML documents may, and should, begin with an XML declaration that specifies the version of XML being used (prolog). In the case of gama-local, the XML input data are followed by the XML document type declaration:     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.3 Tags <gama-local> and <network> A pair tag <gama-local> contains a single pair tag <network> that contains the network definition. The definition of the network is composed of three sections: • <description> of the network (annotation or comments), • network <parameters /> and • <points-observations> section. The sections <description> and <parameters /> are optional, the section <points-observations> 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 <network> 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 <network> tags (analysis of deformations etc.) and definitions of new tags.  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.4 Network description The description of a geodetic network is enclosed in the start-end pair tags <description>. 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 ...   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.5 Network parameters The network parameters may be listed with the following optional attributes of an empty-element tag <parameters /> • 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 (see section Test on 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. 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     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.6 Points and observations The points and observations section is bounded by the pair tag <points-observations> 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 <points-observations> allow for the definition of default values of standard deviations corresponding to observed directions, angles, and distances. • direction-stdev = "…" defines the implicit value of observed direction (default value is not defined) • angle-stdev = "…" defines the implicit value of observed angle (default value is not defined) • zenith-angle-stdev = "…" defines the implicit value of observed zenith angle (default value is not defined) • azimuth-stdev = "…" defines the implicit value of observed azimuth angle (default value is not defined) • distance-stdev = "…" defines the implicit value of observed horizontal distance (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     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.7 Points Points are described by the empty-element tags <point/> 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     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.8 Set of observations The pair tag <obs> 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 <direction … /> • horizontal distances <distance … /> • horizontal angles <angle … /> • slope distances <s-distance … /> • zenith angles <z-angle … /> • azimuths <azimuth … /> • height differences <dh /> The band variance-covariance matrix of directions, distances, angles or other observations listed in one <obs> section may be supplied using a <cov-mat> 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 <cov-mat> and </cov-mat> 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 <obs> 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 <obs> 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 <obs> or <height-differences> section. If entered in the <obs> section, the dist="..." parameter is ignored (Height differences). ## Example   100.00 100.00 100.00 25.00   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.9 Directions Directions are expressed with the following attributes in an empty-element tag <direction /> • to = "…" target point identification • val = "…" observed direction; see section 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 <points-observation direction-stdev="…" > or with a variance-covariance matrix for the given observation set. A similar approach applies to all the observations (distances, angles, etc.) ## Example     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.10 Horizontal distances Distances are written using an empty-element tag <distance /> 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 (<obs>) 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     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.11 Angles Observed angles are expressed with the following attributes of an empty-element tag <angle /> • from = "…" standpoint identification (optional) • bs = "…" backsight target identification • fs = "…" foresight target identification • val = "…" observed angle; see section 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     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.12 Slope distances Slope distances (space distances) are written using an empty-element tag <s-distance /> 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     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.13 Zenith angles Zenith angles are written using an empty-element tag <z-angle /> with the following attributes • from = "…" standpoint identification (optional) • to = "…" target identification • val = "…" observed zenith angle; see section 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     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 <parameters /> tag. Azimuths are expressed with the following attributes in an empty-element tag <azimuth /> • from = "…" standpoint identification • to = "…" target point identification • val = "…" observed azimuth; see section 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 <points-observation azimuth-stdev="…" > or with a variance-covariance matrix for the given observation set. ## Example     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.15 Height differences A set of observed leveling height differences is described using the start-end tag <height-differences> without parameters. The <height-differences> 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 <dh /> 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     [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 2.16 Control coordinates Control (known) coordinates are described by the start-end pair tag <coordinates>. A series of points with known coordinates can be defined using the <point /> tag. The variance-covariance matrix for the entire set of points can be created with a single <cov-mat> tag. In the <point /> tags, a point identification (ID) and its coordinates (x, y and z) must be listed. Although the order of the <point /> 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 <coordinates> tag or outside of it. ## Example   ...   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 <vec> for describing instrument and/or target height are • from_dh = "…" instrument height • to_dh = "…" target height ## Example   ... ..   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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.   XML input stream of points and observation data for program Gama   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] # 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 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 

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## 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).  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 <parameters /> (see section Network parameters) and also gama-local command line parameters (see section 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 (see section Test on 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 Angular units. • latitude is mean latitude in network area. Default value is 50 (gons). • ellipsoid is name of ellipsoid (see section Supported ellipsoids). All fields are mandatory except ellipsoid field. For additional information about handling geodetic systems in gama-local see Tags <gama-local> and <network>. 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, ...   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 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|   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 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 Set of observations. Example (table contents):  conf_id|ccluster|dim|band|tag ----------------------------- 1 |1 |3 |0 |obs 1 |4 |4 |0 |obs   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 tags 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   [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 Control coordinates.  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 Coordinate differences (vectors).  [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 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 Example of local geodetic network to a new file. Note that file should start with <?xml version="1.0" ?> (no whitespace). Alternatively you can use existing XML file from collection of sample networks (see 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 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.

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# 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.

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## 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 

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## 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 

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## 4.3 Parameters of statistical analysis

Program gama-local uses two basic statistical parameters

• confidence probability P (default value is 95%, see parameter conf-pr) and
• actual type of reference standard deviation m0_a (parameter typ-m0).

Confidence probability determines significance level on which statistical tests of adjusted quatities are carried. Actual type of reference standard deviations m0_a specifies whether during statistical analysis we use a priori reference standard deviation m0 or a posteriori estimate m0’.

We can choose only the type of actual reference standard deviation (m0 or m0’) but not its value. The value corresponds to a priori given value of reference standard deviation or to the results of adjustment. 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 used in the 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 quuantity 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.

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## 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 

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## 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 (Parameters of 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 Parameters of statistical analysis.

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 

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## 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 

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## 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).

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## 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. 

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## 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 <parameters update-constrained-coordinates = "yes" /> (see section 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.

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# 5. Data structures and algorithms

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## 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.

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 <typename Observation>
class ObservationData
{
public:
ClusterList<Observation>  CL;

ObservationData();
ObservationData(const ObservationData& cod);
~ObservationData();

ObservationData& operator=(const ObservationData& cod);
template <typename P> void for_each(const P& p) const;
};

template <typename Observation>
class Cluster
{
public:
const ObservationData<Observation>*     observation_data;
ObservationList<Observation>            observation_list;
typename Observation::CovarianceMatrix  covariance_matrix;

Cluster(const ObservationData<Observation>* od);
virtual ~Cluster();

virtual Cluster* clone(const ObservationData<Observation>*) 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 <typename Point>
class PointBase
{
typedef std::map<typename Point::Name, Point*>  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.

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## 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. 1965 [4] andrae1876 6377104.43 300.0 Andrae 1876 (Denmark, Iceland) [4] australian 6378160 298.25 Australian National 1965 [3] bessel 6377397.15508 6356078.96290 Bessel ellipsoid 1841 [1] bessel_nam 6377483.865 299.1528128 Bessel 1841 (Namibia) [4] clarke1858a 6378361 6356685 Clarke ellipsoid 1858 1st [3] clarke1858b 6378558 6355810 Clarke ellipsoid 1858 2nd [3] clarke1866 6378206.4 6356583.8 Clarke ellipsoid 1866 [3] clarke1880 6378316 6356582 Clarke ellipsoid 1880 [3] clarke1880m 6378249.145 293.4663 Clarke ellipsoid 1880 (modified) [4] cpm1799 6375738.7 334.29 Comm. des Poids et Mesures 1799 [4] delambre 6376428 311.5 Delambre 1810 (Belgium) [4] 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 Sarawak) [4] fisher1960 6378166 298.3 Fisher 1960 (Mercury Datum) [3] [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 System 1980 [1] hayford 6378388 297 Hayford 1909 (International) [1] [3] helmert 6378200 298.3 Helmert ellipsoid 1906 [3] hough 6378270 297 Hough [4] iau76 6378140 298.257 IAU 1976 [4] international 6378388 297 International 1924 (Hayford 1909) [1] [3] kaula 6378163 298.24 Kaula 1961 [4] krassovski 6378245 298.3 Krassovski ellipsoid 1940 [1] lerch 6378139 298.257 Lerch 1979 [4] mprts 6397300 191.0 Maupertius 1738 [4] mercury 6378166 298.3 Mercury spheroid 1960 [3] merit 6378137 298.257 MERIT 1983 [4] new_intl 6378157.5 6356772.2 New International 1967 [4] nwl1965 6378145 298.25 Naval Weapons Lab., 1965 [4] plessis 6376523 6355863 Plessis 1817 (France) [4] se_asia 6378155 6356773.3205 Southeast Asia [4] sgs85 6378136 298.257 Soviet Geodetic System 85 [4] schott 6378157 304.5 Schott 1900 spheroid [3] sa1969 6378160 298.25 South American Spheroid 1969 [3] 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 System 1984 [1]
 [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

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## 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(2) that is more effective and sufficiantly accurate and that is used in GNU Gama.

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.

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## 5.4 Class g3::Model

g3::model documentation shall come here ...

namespace GNU_gama {  namespace g3 {

class Model {
public:

typedef GNU_gama::PointBase<g3::Point>              PointBase;
typedef GNU_gama::ObservationData<g3::Observation>  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();
}}


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# A. Copying This Manual

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## A.1 GNU Free Documentation License

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# Concept Index

Jump to: <   A   C   D   F   G   H   I   N   O   P   R   S   T   V   Z
Jump to: <   A   C   D   F   G   H   I   N   O   P   R   S   T   V   Z

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# Footnotes

### (1)

Wikipedia, http://en.wikipedia.org/wiki/Surveying

### (2)

B. R. Bowring: Transformation from spatial to geographical coordinates, Survey Review XXIII, 181, July 1976

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