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public:vievs_manual:global_parameter_estimation

# Global Parameter Estimation

## Introduction

Module Vie_GLOB has been designed and written by Hana Krásná in connection to her Ph.D. thesis. It has the capability to estimate parameters which are common to all VLBI sessions from a so-called global solution, i.e. from a common adjustment of many VLBI sessions. The input data for Vie_GLOB are datum-free normal equations (NEQ) prepared by the module Vie_LSM. The global solution is typically used to determine TRF in terms of station positions and velocities, and the CRF in terms of radio source coordinates.

## Estimated parameters (Version 3.0)

### Parameters estimated from combination of more sessions

station coordinates and velocities: TRF
source coordinates: CRF
Earth orientation parameters
antenna axis offset
station seasonal harmonic signal (annual + semi-annual period)
tidal ERP variations
pole tide Love/Shida number
APL regression coefficients

### Session-wise reduced parameters

zenith wet delay
Earth orientation parameters
station and source coordinates (not suitable for global estimation)

## Input data

For each session there are 3 files: (* denotes name of the session, i.e. 10JAN04XA\_N004)

• *\_an\_glob.mat (in Vie\_GLOB the name of this matlab structure is glob1)a priori station coordinates and velocities used for analysis in this session + information about discontinuities
• *\_Nb\_glob.mat (glob3) datum free normal matrix $N = A^T PA$ and right-hand side vector $b = A^T Pl$
• *\_par\_glob.mat (glob2) glob2.x - information about columns, where the parameters are stored. It also contains estimates from the single session adjustment with Vie_LSM. glob2.opt - information about options, which were chosen for the analysis with VieVS and for preparation of the LEVEL2 data.

Description of the external files needed for Vie\GLOB (VDG = VieVS/DATA/GLOB):

Directory Description
VDG/CRF/DATUM/ arbitraryname.txt files with source names, which will be used for an NNR condition
VDG/CRF/FIXED_SOURCES/ arbitraryname.txt files with source names, which will be fixed to their a priori coordinates (i.e. they will be not involved in the adjustment)
VDG/CRF/REDUCE/ arbitraryname.txt files with source names, which position will be session-wise reduced
VDG/TRF/AO/ arbitraryname.txt files with station names, where the axis offset will be estimated
VDG/TRF/APLRG/ arbitraryname.txt files with station names, where the APL regression coefficients will be estimated
VDG/TRF/DATUM/ arbitraryname.txt files with station names, which will be used for an NNT/NNR condition
VDG/TRF/DISCONT/ arbitraryname.mat files with station names describing the VLBI position discontinuities. The original VLBI-DISCONT.txt file is provided at the web site http://vlbi.geod.uni-bonn.de/IVS-AC/data/VLBI-DISCONT.txt (vlbi\discont.mat is provided within vie\glob and at least one file must be in the ../DISCONT/ directory (also if station coordinates are not estimated))
VDG/TRF/REDUCE/ arbitraryname.txt files with station names, which position will be session-wise reduced. Their velocity will be fixed to the a priori value
VDG/TRF/STSEASON/ arbitraryname.txt files with station names, where the seasonal harmonic signal (annual + semi-annual period) will be estimated
VDG/TRF/VELOC/ arbitraryname.txt files with station names, at which a position discontinuity happened, but we want to estimate a constant velocity for all intervals
VDG/TRF/VELOC/TIES/ arbitraryname.txt files with station names in a special format: names of antennas (8 character for each), the line has to end with “\textbackslash”. Velocity ties will be introduced to stations at one line and the same velocity will be estimated for them.
VieVS/OUT/GLOB/*.m Matlab function backward\solution.m estimates after the global adjustment the session-wise reduced parameters. The plot\backward\*.m functions plot the time series of the session-wise estimated parameters (= output of backward\solution.m)
VieVS/OUT/GLOB/ directories / mat. files are for outputs - data will be automatically written into them

Structure of Vie_GLOB in the VieVS directories:

## Program

The computational strategy of Vie_GLOB follows several steps. First, information from all sessions is read to detect all parameters which are contained in the input normal equations. Only parameters of interest for the global solution are kept in the session-wise NEQ and the remaining parameters are either fixed to their a priori values or reduced from the equations. This can be specified in the GUI, see Figure:

The reduction takes place for parameters which appear in one session only and are dependent on a finite amount of time. These are, for example, the clock parameters, zenith wet delays or tropospheric gradients which can vary by several hours. The reduction means an implicit estimation of such parameters from the session-wise NEQ by a least squares adjustment. The global parameters are detected in the NEQ taken from single sessions, and a new reference number is assigned to each parameter. In the second step of Vie_GLOB the NEQ are reorganized following the new order of parameters (columns/rows in normal matrix and rows in normal vector) and stacked together with the reorganized NEQ from other sessions. This leads to one common global normal matrix which consists of the global parameters only. In the third step conditions (like no-net-rotation and no-net-translation on TRF or no-net-rotation on CRF) and eventually constraints are applied (see Figure below for GUI settings), and by a final inversion of the NEQ system the estimates of the global parameters are obtained. In a usual run of Vie_GLOB where we are only interested in the global parameters (e.g. in a new reference frame) the analysis stops at this stage. However, if we are also interested in the solution for parameters which have been session-wise reduced, a backward solution has to be carried out. This means that the residuals estimated for the global parameters are taken and substituted into the reduced equation going step by step always one level up.

### Reduction and stacking of parameters

The reduction of parameters is based on a division of the normal equation system into two parts. In the first part those parameters are concentrated, which will be kept in the global matrix, and in the second part parameters are ordered, which will be estimated only from a single session:

$$\left[ \begin{array}{cc} N_{11} & N_{12} \\ N_{21} & N_{22} \\ \end{array} \right] \cdot \left[ \begin{array}{cc} dx_{1} \\ dx_{2} \\ \end{array} \right]=\left[ \begin{array}{cc} b_{1} \\ b_{2} \\ \end{array} \right].$$

The matrix equation above corresponds to the following two coupled equations:

$$N_{11} \cdot dx_{1} + N_{12} \cdot dx_{2} = b_{1},$$

$$N_{21} \cdot dx_{1} + N_{22} \cdot dx_{2} = b_{2}.$$

From the equation above vector $dx_{2}$ can be expressed containing the reduced parameters:

$$dx_{2} = N_{22}^{-1} \cdot b_{2} - N_{22}^{-1} N_{21} \cdot dx_{1}$$

and substituted into the equation for $N_{11}$:

$$N_{11} \cdot dx_{1} + N_{12} N_{22}^{-1} \cdot b_{2} - N_{12} N_{22}^{-1} N_{21} \cdot dx_{1} = b_{1} ,$$

$$(N_{11} - N_{12} N_{22}^{-1} N_{21} ) \cdot dx_{1} = b_{1} - N_{12} N_{22}^{-1} \cdot b_{2},$$

$$N_{R} \cdot dx_{1} = b_{R}.$$

The reduced N matrix $N_{R}$ and the reduced b vector $b_{R}$ are than “stacked” with reduced normal equation systems from other sessions and a global N matrix $N_{G}$ and a global b vector $b_{G}$ is created. Attention has to be paid so that the order of parameters is identical in the reduced normal equation systems:

$$N_{G}=N_{R1} + N_{R2}+ N_{R3}+ \cdots ,$$ $$b_{G}=b_{R1} + b_{R2}+ b_{R3}+ \cdots .$$

The final solution for the global parameters is done using an inversion of the global normal equation system:

$$dx_{G}=N_{G}^{-1} \cdot b_{G}.\\$$

The estimates of the session-wise reduced parameters can be obtained by substituting the vector $dx_{G}$ into the equation for $dx_{2}$, where $dx_{1} = dx_{G}$ and thus contains the globally adjusted parameters. It is obvious that one has to store the matrices $N_{22}$, $N_{21}$ and vectors $b_{2}$ of each session. To obtain the time series of the reduced parameters from all sessions it should be started with the reduced normal

## Output data

### Output figures

Output figures are saved in VieVS/OUT/GLOB/_PLOTS/TEST_OUT (source code for saving the figures is at the very end of the main program Vie_GLOB.m).

The figure below shows stations in sessions included in the global adjustment. It is created with the function plot_antactiv.m. It is stored in VieVS/OUT/GLOB/_PLOTS/TEST_OUT/ant_activity_TEST_LEVEL2.eps.

The figure below shows the map with all stations included in the global adjustment. Blue circles denote stations included in the NNT/NNR condition, red circles show stations excluded from the NNT/NNR condition. It is created with the function plot_ant.m. It is stored in VieVS/OUT/GLOB/_PLOTS/TEST_OUT/ant_map_TEST_LEVEL2.eps.

The figure below shows the map with all stations included in the global adjustment. Blue circles denote stations included in the NNT/NNR condition, red circles show stations excluded from the NNT/NNR condition. It is created with the function plot\_ant.m. It is stored in VieVS/OUT/GLOB/_PLOTS/TEST_OUT/ant_map_TEST_LEVEL2.eps:

Thew figure below shows the map with all sources included in the global adjustment. Blue circles denote sources included in the NNR condition, red circles are sources excluded from the NNR condition. Sources which were fixed to their a priori coordinates are not shown. It is created with the function plot_sou.m. It is stored in VieVS/OUT/GLOB/_PLOTS/TEST_OUT/sou_map_TEST_LEVEL2.eps:

### Results

Estimates are stored in VieVS/OUT/GLOB/_ESTIMATES/TEST_OUT/ :

• in Matlab format: globsol_TEST_LEVEL2.mat
• in TXT format: glob_results_TEST_LEVEL2.txt

## global EOP

You can use vie_glob to estimate EOP. The advantage is, that instead of the usually 3 values for the single session solution, the EOP series of the consecutive sessions are combined. This gives you one value per day. Important: if you want to globally estimate EOP, choose loose constraints (e.g. 1 mas) in the single session solution.