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Datum Definition and its Influence on the Sensitivity of Geodetic Monitoring Networks

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and its Influence on the Sensitivity of

Geodetic Monitoring Networks

Gilad Even-Tzur

Department of Mapping and Geo-Information Engineering

Faculty of Civil and Environmental Engineering

TECHNION - Israel Institute of Technology

12th FIG Symposium on Deformation Measurements

May 22 - 24, 2006, Baden, Austria

of Geodetic Monitoring Network

Geodetic monitoring networks should be designed and tested according to three criteria that determine their quality and utility:

• Accuracy -

• Reliability -

• Sensitivity -

The network’s quality in terms of random errors.

The network’s ability to sense and identify gross errors in the measurements.

The ability to detect and measure movements and deformations in the area covered by the network.

► A datum is defined as a subset of network points which remain stable and consistent over time.

► The primary goal of deformation analysis is the determination of velocities of points located on a deformed area. The velocities are unique parameters capable of representing the deformation as four dimensional phenomenon.

► The measurements are used to estimate the coordinates of the network points, for a reference epoch and the velocity of the points, .

► The observation equations for solving and are rank deficient due to the lack of datum. We need to define a reference coordinate systems for and , and thus the datum defect is double.

►The datum defect is corrected by adding a corresponding number of constraints to the unknown parameters.

► The constraints that apply to the network points could be different from those applied to the velocities.

► As we are interested mainly in the velocities, our concern remains with the selection and definition of a meaningful reference system for .

► Velocities are estimated based on a time series of monitoring campaigns. Therefore, we should not obtain the datum definition from the measurements but rather assume a datum defect of 7 or even 12 parameters (translations, rotations, scales and obliquities).

► We use the S-transformation to transform one solution pertaining to a certain datum into another solution pertaining to another datum.

The sensitivity of the network is examined using the statistical test of hypothesis.

The null hypothesis

can be tested against the alternative hypothesis

If the null hypothesis is rejected where:

The Sensitivity of a group of points

rigid motion

homogenous strain parameters

The velocity field of a congruent group of points can be partitioned into a linear model and a residual vector v:

where:

translation rotation scale obliquity

The Sensitivity of a group of points (cont.)

And matrix B has the following composition:

The coordinates of point i are given in a Cartesian system that is parallel to the reference system and with origin at the network’s barycenter.

The Sensitivity of a group of points (cont.)

Where:

By using the linear part of the model for the velocity field we can define the test statistic for a congruent group of points as:

The Sensitivity of a group of points (cont.)

For a group of points can be regarded as invariant, r is a constant, and is the variance factor.

The product defines the sensitivity of the group of points.

The larger the trace of is, the more sensitive the network is, or:the more accurate g is, the more sensitive the network is.

The optimal geometrical distribution of points

The accuracy of g depends on the accuracy of the velocity field.

In order to determine the best ability for monitoring individual components of vector g we can examine the optimal geometrical distribution of points according to the sensitivity criteria based on the product.

The optimal geometrical distribution of points

Translation - A single point can define the velocity of a certain area covered by the network. So the ability to determine the velocity of the barycenter of one of the network blocks is not influenced by the geometrical distribution of points.

The optimal geometrical distribution of points

X

Y

Rotation - Two points are needed to determine the rotation of a block. The more distant these points are, the better the rotation can be detected.

The optimal geometrical distribution of points

X

Y

Scale Change- Two points, A and B for example, with

or , can not distinguish a difference in the X or Y scale. We aim for a coordinate difference between points that will be as large as possible in both components.

The optimal geometrical distribution of points

X

Y

Variation of angle between axes- Three points are needed to detect the angle between the two axes in a two-dimensional network. The optimal distribution of points is when the distance in both coordinate components is as large as possible.

Datum definition and its influence on sensitivity

Let us investigate the influence of datum definition on the sensitivity of the network, and explore the relationship between the datum points’ geometry and the sensitivity of the network points.

Assumptions for simplification:

► The geometrical distribution of each group is identical relative to the fault line.

► Each group of points can serve as a datum, meaning that the relative position of each group of points is invariant in time.

► The covariance matrix of the velocities remains the same when datum transformation is implemented between blocks.

Datum definition and its influence on sensitivity

fault line

Block 1

Block 2

Let us investigate which group of points should define the datum when the objective is achieving a sensitive network.

Datum definition and its influence on sensitivity

fault line

Reference Block

Object Block

In this situation the sensitivity of the network can be increased if the geometrical distribution of the object points will be wide.

Movements and deformation of the object block are defined in relation to the reference block.

When translation is considered between the reference block and the object block:

Object Block

Reference Block

The sensitivity is not dependent on the reference block chosen, and each block can serve as a datum where the network sensitivity is not influenced by the geometrical distribution of points.

Reference Block

Object Block

+

barycenter

The new object block rotates in the opposite direction to the original rotation at the same rate, and has a translation as well.

The new object block rotates in the opposite direction to the original rotation at the same rate,

The velocities are greater and the test statistic t increases.

As long as the barycenter of the reference block is close to the rotation axis, the sensitivity of the object points increases.

In this situation the sensitivity of the network increases as the distance between the object points increases.

When scale change or obliquity are considered:

The new object block becomes deformed in an opposite rate and also has a translation.

The velocities of the new object block are greater and the test statistic t increases.

As long as points on the deformed block are used to define the datum, the sensitivity of the object points increases.

In this situation the sensitivity of the network increases when the coordinate difference between points is as large as possible in both components, and the distance in both coordinate components is as large as possible.

► The geometric distribution of the datum points does not dramatically affect the sensitivity of the network.

► The geometric distribution of the object points affect the sensitivity of the network.

► As strange as it may seem, it is important to establish the datum based on points that represent the deformed block.

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