Evsphinx.addnodesdocument)}( rawsourcechildren](docutils.nodessubstitution_definition)}(h.. |rel| replace:: 10.71h]h Text10.71}parenth sba attributes}(ids]classes]names]reladupnames]backrefs]utagnameh source lineKhh _documenthubh )}(h.. |date| replace:: |today| h]h2024 136}(hh)h(hh%Nh'Nubah}(h]h]h]dateah ]h"]uh$h h%h&h'Khhh(hubh target)}(h.. _intro_gnss_solve:h]h}(h]h]h]h ]h"]refidintro-gnss-solveuh$h8h'Khhh(hh%D/chandler/home/floyd/public_html/gg/docs/source/intro/gnss/solve.rstubh section)}(hhh](h title)}(hParameter Estimationh]hParameter Estimation}(hhNh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hLhhIh(hh%hFh'Kubh paragraph)}(hXeGAMIT (program :program:`solve`) incorporates a weighted least squares algorithm to estimate the relative positions of a set of stations, orbital and Earth-rotation parameters, zenith delays, and phase ambiguities by fitting to doubly differenced phase observations. Since the functional (mathematical) model relating the observations and parameters is non-linear, GAMIT produces two solutions, the first to obtain coordinates within a few decimeters, and the second to obtain the final estimates. (See the discussion in Section 2.2 of the `GAMIT Reference Manual `_.)h](hGAMIT (program }(hh^h(hh%Nh'Nubhliteral_strong)}(h:program:`solve`h]hsolve}(hhhh(hh%Nh'Nubah}(h]h]programah]h ]h"]uh$hfhh^ubhX) incorporates a weighted least squares algorithm to estimate the relative positions of a set of stations, orbital and Earth-rotation parameters, zenith delays, and phase ambiguities by fitting to doubly differenced phase observations. Since the functional (mathematical) model relating the observations and parameters is non-linear, GAMIT produces two solutions, the first to obtain coordinates within a few decimeters, and the second to obtain the final estimates. (See the discussion in Section 2.2 of the }(hh^h(hh%Nh'Nubh reference)}(hG`GAMIT Reference Manual `_h]hGAMIT Reference Manual}(hh}h(hh%Nh'Nubah}(h]h]h]h ]h"]nameGAMIT Reference Manualrefuri+http://geoweb.mit.edu/gg/docs/GAMIT_Ref.pdfuh$h{hh^ubh9)}(h. h]h}(h]gamit-reference-manualah]h]gamit reference manualah ]h"]refurihuh$h8 referencedKhh^ubh.)}(hh^h(hh%Nh'Nubeh}(h]h]h]h ]h"]uh$h\h%hFh'KhhIh(hubh])}(hX-In current practice, the GAMIT solution is not usually used directly to obtain the final estimates of station positions from a survey. Rather, we use GAMIT to produce estimates and an associated covariance matrix ("quasi-observations") of station positions and (optionally) orbital and Earth-rotation parameters which are then input to GLOBK or other similar programs to combine the data with those from other networks and times to estimate positions and velocities :cite:p:`Feigl_et_al_1993,Dong_et_al_1998`. GLOBK uses a Kalman filter (equivalent to sequential least squares if there are no stochastic parameters in the solution) which operates on covariance matrices rather than normal equations and hence requires that you specify a non-infinite *a priori* constraint for each parameter estimated (see, e.g., :cite:t:`Herring_et_al_1990`). In order not to bias the combination, GAMIT generates the solution used by GLOBK with loose constraints on the parameters. Since phase ambiguities must be resolved (if possible) in the phase processing, however, GAMIT also generates several intermediate solutions with user-defined constraints before loosening the constraints for its final solution. These steps are described in detail in Section 3.4 of the `GAMIT Reference Manual `_.h](hXIn current practice, the GAMIT solution is not usually used directly to obtain the final estimates of station positions from a survey. Rather, we use GAMIT to produce estimates and an associated covariance matrix (“quasi-observations”) of station positions and (optionally) orbital and Earth-rotation parameters which are then input to GLOBK or other similar programs to combine the data with those from other networks and times to estimate positions and velocities }(hhh(hh%Nh'Nubh pending_xref)}(h*:cite:p:`Feigl_et_al_1993,Dong_et_al_1998`h]h inline)}(hhh]h Feigl_et_al_1993,Dong_et_al_1998}(hhh(hh%Nh'Nubah}(h]h](xrefcitecite-peh]h ]h"]uh$hhhubah}(h]id1ah]h]h ]h"]refdocintro/gnss/solve refdomainhŒreftypep refexplicitrefwarn reftarget Feigl_et_al_1993,Dong_et_al_1998uh$hh%hFh'K hhubh. GLOBK uses a Kalman filter (equivalent to sequential least squares if there are no stochastic parameters in the solution) which operates on covariance matrices rather than normal equations and hence requires that you specify a non-infinite }(hhh(hh%Nh'Nubh emphasis)}(h *a priori*h]ha priori}(hhh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhhubh5 constraint for each parameter estimated (see, e.g., }(hhh(hh%Nh'Nubh)}(h:cite:t:`Herring_et_al_1990`h]h)}(hhh]hHerring_et_al_1990}(hhh(hh%Nh'Nubah}(h]h](hcitecite-teh]h ]h"]uh$hhhubah}(h]id2ah]h]h ]h"]refdochό refdomainhreftypet refexplicitrefwarnhՌHerring_et_al_1990uh$hh%hFh'K hhubhX). In order not to bias the combination, GAMIT generates the solution used by GLOBK with loose constraints on the parameters. Since phase ambiguities must be resolved (if possible) in the phase processing, however, GAMIT also generates several intermediate solutions with user-defined constraints before loosening the constraints for its final solution. These steps are described in detail in Section 3.4 of the }(hhh(hh%Nh'Nubh|)}(hG`GAMIT Reference Manual `_h]hGAMIT Reference Manual}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]nameGAMIT Reference Manualh+http://geoweb.mit.edu/gg/docs/GAMIT_Ref.pdfuh$h{hhubh9)}(h. h]h}(h]id3ah]h]h ]gamit reference manualah"]refurij$uh$h8hKhhubh.}(hhh(hh%Nh'Nubeh}(h]h]h]h ]h"]uh$h\h%hFh'K hhIh(hubh])}(hXIn parameter estimation based on least-squares, the conventional measure of goodness-of-fit is the :math:`\chi^2` (chi-square) statistic, defined for uncorrelated data as the sum of the squares of each observation residual (post-fit observed minus computed observation, ":math:`\text{o} - \text{c}`") divided by its assigned uncertainty. In a GPS analysis parameter correlations arise (even if not represented in the original data weights) so the computation of :math:`\chi^2` in GAMIT or GLOBK involves a complex matrix operation (see :cite:t:`Dong_et_al_1998`), but the idea is the same. The value of :math:`\chi^2` is usually normalized by dividing by the "degrees of freedom" (:math:`df`), the number of observations minus the number of parameters estimated, so that the ideal value for properly weighted, independent random observations is 1.0. Conversely, with whatever *a priori* weight is assigned to the observations, multiplying the estimated parameter uncertainties by the square root of :math:`\chi^2/df` yields the "formal errors" ("formal standard deviations") of the parameter. With white noise, these formal uncertainties will be realistic and will be inversely proportional to the square root of the number of observations (i.e., will depend strongly on the sampling interval).h](hcIn parameter estimation based on least-squares, the conventional measure of goodness-of-fit is the }(hj<h(hh%Nh'Nubh math)}(h:math:`\chi^2`h]h\chi^2}(hjFh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhj<ubh (chi-square) statistic, defined for uncorrelated data as the sum of the squares of each observation residual (post-fit observed minus computed observation, “}(hj<h(hh%Nh'NubjE)}(h:math:`\text{o} - \text{c}`h]h\text{o} - \text{c}}(hjXh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhj<ubh”) divided by its assigned uncertainty. In a GPS analysis parameter correlations arise (even if not represented in the original data weights) so the computation of }(hj<h(hh%Nh'NubjE)}(h:math:`\chi^2`h]h\chi^2}(hjjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhj<ubh< in GAMIT or GLOBK involves a complex matrix operation (see }(hj<h(hh%Nh'Nubh)}(h:cite:t:`Dong_et_al_1998`h]h)}(hj~h]hDong_et_al_1998}(hjh(hh%Nh'Nubah}(h]h](hcitecite-teh]h ]h"]uh$hhj|ubah}(h]id4ah]h]h ]h"]refdochό refdomainjreftypej  refexplicitrefwarnhՌDong_et_al_1998uh$hh%hFh'Khj<ubh*), but the idea is the same. The value of }(hj<h(hh%Nh'NubjE)}(h:math:`\chi^2`h]h\chi^2}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhj<ubhD is usually normalized by dividing by the “degrees of freedom” (}(hj<h(hh%Nh'NubjE)}(h :math:`df`h]hdf}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhj<ubh), the number of observations minus the number of parameters estimated, so that the ideal value for properly weighted, independent random observations is 1.0. Conversely, with whatever }(hj<h(hh%Nh'Nubh)}(h *a priori*h]ha priori}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhj<ubhq weight is assigned to the observations, multiplying the estimated parameter uncertainties by the square root of }(hj<h(hh%Nh'NubjE)}(h:math:`\chi^2/df`h]h \chi^2/df}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhj<ubhX yields the “formal errors” (“formal standard deviations”) of the parameter. With white noise, these formal uncertainties will be realistic and will be inversely proportional to the square root of the number of observations (i.e., will depend strongly on the sampling interval).}(hj<h(hh%Nh'Nubeh}(h]h]h]h ]h"]uh$h\h%hFh'KhhIh(hubh])}(hXThe uncertainties in a GNSS analysis, however, cannot be treated with white noise statistics because errors with temporal correlations dominate both the phase observations and estimates of station coordinates (the quasi-observations input to GLOBK). In the phase residuals (from cview or the sky plots produced by :program:`sh_gamit`), the visible noise—from multipathing and tropospheric fluctuations—is typically correlated over spans of 15–30 minutes. This implies that only samples taken at these intervals are independent, and, to a first approximation, we would get realistic uncertainties by multiplying the formal errors by the square root of the ratio of this interval to the sampling interval used—e.g., for the 2 minute sampling commonly used in :program:`solve`, we would increase the uncertainties by a factor of 3–5. There also errors with longer correlation times that do not show up in the residuals but are absorbed into the parameter adjustments. Assessing the magnitude of these errors requires us to use noise visible in the residuals (phase or coordinates) to infer the character of the noise at lower frequencies. There are a number of excellent studies of the character of the errors discussed in the presentation "Error_Analysis.ppt" from our most recent workshops on the Documentation page of the GAMIT/GLOBK web site (e.g., :cite:t:`Mao_et_al_1999`, :cite:t:`Dixon_et_al_2000`, and :cite:t:`Williams_2003`). Once you have adopted a particular weighting of the data, it is often possible to use external knowledge of the expected behavior of the coordinates or velocities to validate the uncertainties; see :cite:t:`McClusky_et_al_2000`, :cite:t:`Davis_et_al_2003`, and :cite:t:`McCaffrey_et_al_2007`.h](hX:The uncertainties in a GNSS analysis, however, cannot be treated with white noise statistics because errors with temporal correlations dominate both the phase observations and estimates of station coordinates (the quasi-observations input to GLOBK). In the phase residuals (from cview or the sky plots produced by }(hjh(hh%Nh'Nubhg)}(h:program:`sh_gamit`h]hsh_gamit}(hjh(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubhX), the visible noise—from multipathing and tropospheric fluctuations—is typically correlated over spans of 15–30 minutes. This implies that only samples taken at these intervals are independent, and, to a first approximation, we would get realistic uncertainties by multiplying the formal errors by the square root of the ratio of this interval to the sampling interval used—e.g., for the 2 minute sampling commonly used in }(hjh(hh%Nh'Nubhg)}(h:program:`solve`h]hsolve}(hjh(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubhXG, we would increase the uncertainties by a factor of 3–5. There also errors with longer correlation times that do not show up in the residuals but are absorbed into the parameter adjustments. Assessing the magnitude of these errors requires us to use noise visible in the residuals (phase or coordinates) to infer the character of the noise at lower frequencies. There are a number of excellent studies of the character of the errors discussed in the presentation “Error_Analysis.ppt” from our most recent workshops on the Documentation page of the GAMIT/GLOBK web site (e.g., }(hjh(hh%Nh'Nubh)}(h:cite:t:`Mao_et_al_1999`h]h)}(hjh]hMao_et_al_1999}(hjh(hh%Nh'Nubah}(h]h](hcitecite-teh]h ]h"]uh$hhjubah}(h]id5ah]h]h ]h"]refdochό refdomainj(reftypej  refexplicitrefwarnhՌMao_et_al_1999uh$hh%hFh'Khjubh, }(hjh(hh%Nh'Nubh)}(h:cite:t:`Dixon_et_al_2000`h]h)}(hj@h]hDixon_et_al_2000}(hjBh(hh%Nh'Nubah}(h]h](hcitecite-teh]h ]h"]uh$hhj>ubah}(h]id6ah]h]h ]h"]refdochό refdomainjLreftypej  refexplicitrefwarnhՌDixon_et_al_2000uh$hh%hFh'Khjubh, and }(hjh(hh%Nh'Nubh)}(h:cite:t:`Williams_2003`h]h)}(hjdh]h Williams_2003}(hjfh(hh%Nh'Nubah}(h]h](hcitecite-teh]h ]h"]uh$hhjbubah}(h]id7ah]h]h ]h"]refdochό refdomainjpreftypej  refexplicitrefwarnhՌ Williams_2003uh$hh%hFh'Khjubh). Once you have adopted a particular weighting of the data, it is often possible to use external knowledge of the expected behavior of the coordinates or velocities to validate the uncertainties; see }(hjh(hh%Nh'Nubh)}(h:cite:t:`McClusky_et_al_2000`h]h)}(hjh]hMcClusky_et_al_2000}(hjh(hh%Nh'Nubah}(h]h](hcitecite-teh]h ]h"]uh$hhjubah}(h]id8ah]h]h ]h"]refdochό refdomainjreftypej  refexplicitrefwarnhՌMcClusky_et_al_2000uh$hh%hFh'Khjubh, }hjsbh)}(h:cite:t:`Davis_et_al_2003`h]h)}(hjh]hDavis_et_al_2003}(hjh(hh%Nh'Nubah}(h]h](hcitecite-teh]h ]h"]uh$hhjubah}(h]id9ah]h]h ]h"]refdochό refdomainjreftypej  refexplicitrefwarnhՌDavis_et_al_2003uh$hh%hFh'Khjubh, and }hjsbh)}(h:cite:t:`McCaffrey_et_al_2007`h]h)}(hjh]hMcCaffrey_et_al_2007}(hjh(hh%Nh'Nubah}(h]h](hcitecite-teh]h ]h"]uh$hhjubah}(h]id10ah]h]h ]h"]refdochό refdomainjreftypej  refexplicitrefwarnhՌMcCaffrey_et_al_2007uh$hh%hFh'Khjubh.}(hjh(hh%Nh'Nubeh}(h]h]h]h ]h"]uh$h\h%hFh'KhhIh(hubh])}(hX$In GAMIT/GLOBK there are several ways you can control the uncertainties you obtain for coordinates and velocities, and it is important for you to keep clearly in mind how each of these operates. The uncertainties generated by :program:`solve` and passed to GLOBK in the h-file are determined by the *a priori* error assigned to the phase observations and by the sampling interval—:program:`solve` does *not* rescale by the square root of :math:`\chi^2/df` ("postfit nrms" in the q-file). In the initial ("preliminary") solution, we normally assign an uncertainty of 10 mm to each one-way L1 phase. By Equation :eq:`eqn:lc_phase`, the assigned uncertainty in an LC phase becomes 32 mm. The mean rms of one-way LC residuals is typically ~ 6–9 mm, so the nrms computed by :program:`solve` is 0.2–0.3. In the second ("final") solution, we normally reweight the observations using a constant and elevation-dependent term computed in data editing by program :program:`autcln` from the actual (one-way LC) phase residuals. In order to keep the overall weighting approximately the same as with the 10 mm constant error, the values computed by :program:`autcln` (:content:`ATELV` table in file :file:`autcln.post.sum`) are multiplied by an arbitrary factor of 1.7 (in script :program:`sh_sigelv`) before being input to :program:`solve` (via the N-file). We chose to use inflated values of the *a priori* phase error and not rescale by the nrms in order to generate coordinate uncertainties that (in the presence of correlated noise) are approximately realistic with 2-minute sampling. An equally valid approach would be to rescale by the nrms (i.e. make :math:`\chi^2/df = 1.0`) and compensate later for the unrealistically low coordinate uncertainties. With whatever weighting you use in :program:`solve`, you can increase the coordinate uncertainties used by GLOBK by rescaling all covariances on the h-file or by adding white noise or random-walk noise to the variances of individual stations. We discuss in :numref:`intro_prod_weighting` why the latter approach is usually preferred.h](hIn GAMIT/GLOBK there are several ways you can control the uncertainties you obtain for coordinates and velocities, and it is important for you to keep clearly in mind how each of these operates. The uncertainties generated by }(hjh(hh%Nh'Nubhg)}(h:program:`solve`h]hsolve}(hjh(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubh9 and passed to GLOBK in the h-file are determined by the }(hjh(hh%Nh'Nubh)}(h *a priori*h]ha priori}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhjubhI error assigned to the phase observations and by the sampling interval—}(hjh(hh%Nh'Nubhg)}(h:program:`solve`h]hsolve}(hj$h(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubh does }(hjh(hh%Nh'Nubh)}(h*not*h]hnot}(hj6h(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhjubh rescale by the square root of }(hjh(hh%Nh'NubjE)}(h:math:`\chi^2/df`h]h \chi^2/df}(hjHh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhjubh (“postfit nrms” in the q-file). In the initial (“preliminary”) solution, we normally assign an uncertainty of 10 mm to each one-way L1 phase. By Equation }(hjh(hh%Nh'Nubh)}(h:eq:`eqn:lc_phase`h]h literal)}(hj\h]h eqn:lc_phase}(hj`h(hh%Nh'Nubah}(h]h](heqeh]h ]h"]uh$j^hjZubah}(h]h]h]h ]h"]refdochό refdomainjDreftypejj refexplicitrefwarnhՌ eqn:lc_phaseuh$hh%hFh'Khjubh, the assigned uncertainty in an LC phase becomes 32 mm. The mean rms of one-way LC residuals is typically ~ 6–9 mm, so the nrms computed by }(hjh(hh%Nh'Nubhg)}(h:program:`solve`h]hsolve}(hj~h(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubh is 0.2–0.3. In the second (“final”) solution, we normally reweight the observations using a constant and elevation-dependent term computed in data editing by program }(hjh(hh%Nh'Nubhg)}(h:program:`autcln`h]hautcln}(hjh(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubh from the actual (one-way LC) phase residuals. In order to keep the overall weighting approximately the same as with the 10 mm constant error, the values computed by }(hjh(hh%Nh'Nubhg)}(h:program:`autcln`h]hautcln}(hjh(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubh (}(hjh(hh%Nh'Nubj_)}(h:content:`ATELV`h]hATELV}(hjh(hh%Nh'Nubah}(h]h](code highlightcontenttexteh]h ]h"]languagejuh$j^hjubh table in file }(hjh(hh%Nh'Nubj_)}(h:file:`autcln.post.sum`h]hautcln.post.sum}(hjh(hh%Nh'Nubah}(h]h]fileah]h ]h"]rolefileuh$j^hjubh:) are multiplied by an arbitrary factor of 1.7 (in script }(hjh(hh%Nh'Nubhg)}(h:program:`sh_sigelv`h]h sh_sigelv}(hjh(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubh) before being input to }(hjh(hh%Nh'Nubhg)}(h:program:`solve`h]hsolve}(hjh(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubh: (via the N-file). We chose to use inflated values of the }(hjh(hh%Nh'Nubh)}(h *a priori*h]ha priori}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhjubh phase error and not rescale by the nrms in order to generate coordinate uncertainties that (in the presence of correlated noise) are approximately realistic with 2-minute sampling. An equally valid approach would be to rescale by the nrms (i.e. make }(hjh(hh%Nh'NubjE)}(h:math:`\chi^2/df = 1.0`h]h\chi^2/df = 1.0}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhjubhp) and compensate later for the unrealistically low coordinate uncertainties. With whatever weighting you use in }(hjh(hh%Nh'Nubhg)}(h:program:`solve`h]hsolve}(hj(h(hh%Nh'Nubah}(h]h]hsah]h ]h"]uh$hfhjubh, you can increase the coordinate uncertainties used by GLOBK by rescaling all covariances on the h-file or by adding white noise or random-walk noise to the variances of individual stations. We discuss in }(hjh(hh%Nh'Nubh)}(h:numref:`intro_prod_weighting`h]j_)}(hj<h]hintro_prod_weighting}(hj>h(hh%Nh'Nubah}(h]h](hstd std-numrefeh]h ]h"]uh$j^hj:ubah}(h]h]h]h ]h"]refdochό refdomainjHreftypenumref refexplicitrefwarnhՌintro_prod_weightinguh$hh%hFh'Khjubh. why the latter approach is usually preferred.}(hjh(hh%Nh'Nubeh}(h]h]h]h ]h"]uh$h\h%hFh'KhhIh(hubh])}(hXTo obtain meaningful estimates of crustal motion, it is necessary to define a reference frame by imposing constraints on the solution. These come in two common flavors: With *finite* constraints, we assign realistic *a priori* uncertainties to the coordinates and velocities of one or more sites. This is the only option available in GAMIT and it is also available in GLOBK. A minimum (non-redundant) constraint with this approach would involve restricting translation by fixing or tightly constraining the three coordinates of one site, and restricting rotation by constraining Earth orientation. Constraining additional sites provides redundancy, but can distort the network and remove your ability to detect errors in those sites except through an increase in :math:`\chi^2`. The second type of constraint, available through the glorg program of GLOBK, is *generalized*. With this approach, we choose as large a set as possible of sites with good *a priori* values and minimize their adjustments while estimating an overall translation, rotation, and scale (Helmert parameters) of the network. Since all of the frame-defining sites are free to move, outliers can be readily detected and removed. Moreover, with generalized constraints there can be no internal distortion of the network: all realizations of the reference frame with a given data set will differ only by a translation and rotation. See :cite:t:`Dong_et_al_1998` for a mathematical description of each of these approaches.h](hTo obtain meaningful estimates of crustal motion, it is necessary to define a reference frame by imposing constraints on the solution. These come in two common flavors: With }(hjdh(hh%Nh'Nubh)}(h*finite*h]hfinite}(hjlh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhjdubh" constraints, we assign realistic }(hjdh(hh%Nh'Nubh)}(h *a priori*h]ha priori}(hj~h(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhjdubhX uncertainties to the coordinates and velocities of one or more sites. This is the only option available in GAMIT and it is also available in GLOBK. A minimum (non-redundant) constraint with this approach would involve restricting translation by fixing or tightly constraining the three coordinates of one site, and restricting rotation by constraining Earth orientation. Constraining additional sites provides redundancy, but can distort the network and remove your ability to detect errors in those sites except through an increase in }(hjdh(hh%Nh'NubjE)}(h:math:`\chi^2`h]h\chi^2}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$jDhjdubhR. The second type of constraint, available through the glorg program of GLOBK, is }(hjdh(hh%Nh'Nubh)}(h *generalized*h]h generalized}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhjdubhN. With this approach, we choose as large a set as possible of sites with good }(hjdh(hh%Nh'Nubh)}(h *a priori*h]ha priori}(hjh(hh%Nh'Nubah}(h]h]h]h ]h"]uh$hhjdubhX values and minimize their adjustments while estimating an overall translation, rotation, and scale (Helmert parameters) of the network. Since all of the frame-defining sites are free to move, outliers can be readily detected and removed. Moreover, with generalized constraints there can be no internal distortion of the network: all realizations of the reference frame with a given data set will differ only by a translation and rotation. 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