Table Of Contents

• Introduction To Testing Survey Network Adjustments
• Factors Affecting Adjustments
  • The Mathematical Model
  • The Stochastic Model
  • Gross Errors
• Confidence Levels
• Redundant Measurements
• Specific Tests
  • Estimate of the Variance Factor (Global Test)
  • Residuals (Local Test)
• Testing Procedure
• Accuracy
• Reliability

Introduction To Testing Survey Network Adjustments

Least squares adjustments of survey networks rarely give acceptable adjustment results immediately due to a number of different factors. One or more of the factors are likely to be present unless a standard adjustment technique is applied to measurements acquired by well tried procedures and experienced observers.

Therefore adjustment results must be tested to detect and/or eliminate any extraneous influences. The testing of adjustments may be applied to any least squares solution of any type(s) of measurements, but find their most frequent use in testing the adjustment of survey networks.

back to top

Factors Affecting Adjustments

The Mathematical Model

The general term "mathematical model" refers to the equations used to emulate the physical situation, or in other words, the mathematical relationship between the measurements and the unknown parameters. It also refers to the number and type of unknowns carried in the adjustment to describe the physical aspects of the measurements.

Mathematical models may be:

• Inaccurate, inappropriate or just plain wrong.

• Under-parameterised

  There are insufficient unknown parameters to account for the physical aspects of measurement. Such a mathematical model is said to have unmodelled systematic errors. A good example is additive constants or scale errors in an EDM which have not been calibrated or the effects have not been removed prior to the adjustment.

• Over-parameterised

  There are too many unknown parameters in the adjustment and some have no physical reality or function. An example is carrying EDM errors in a survey adjustment where none exist or they have been removed prior to the adjustment.

back to top

The Stochastic Model

The general term "stochastic model" refers to the statistical properties of the measurements, as described by the weight coefficient matrix used in the adjustment. This includes the relationships between the measurement types and their precision (constant, linear, ...), the magnitudes of the precisions, the relative weights for different measurement types, and the presence or absence of correlation terms.

Stochastic models may be incorrect in any of the above aspects, a typical example being the neglect of correlations between horizontal angles. Common problems with untested measurement techniques, unexperienced observers or simply unfamiliar situations are:

• Under-estimation of precisions

  The measurements are made to a better precision than that expected.

• Over-estimation of precisions

  The measurements are made to a worse precision than that expected.

back to top

Gross Errors

Gross errors, blunders or outliers are caused by human mistakes in measurements, reductions, transcriptions, and equipment errors or anomalous physical circumstances. Gross errors are an integral part of survey adjustments and are a statistical certainty! This is because measurements are assumed to follow a normal distribution, which implies that there is no theoretical limit to how far an individual value may depart from the mean.

back to top

Confidence Levels

The basis for all testing of adjustments is the specification of confidence levels or limits. Levels are specified in terms of probabilities which can be related the departure of a value from the mean by the distribution function. A confidence level of 99.9% implies that 999 times out of 1000 acceptable measurements are made, whilst 1 time in 1000 an unacceptable measurement or gross error is made, which is rejected.

If the probability function of the measurements is known their limits can be set based on the departure from the mean, by "cutting off" the area under the probability density curve. The area inside the cut-offs or limits is set by the probability level. The limits are conveniently expressed in terms of standard deviations of the measurement.

The "rule of thumb" of three standard deviations (3s) for measurement rejection corresponds to a confidence level of 99.75% for a normally distributed measurement:

In effect the rule of thumb says that 25 times in 10000 measurements a gross error is expected. (m - 3s) and (m + 3s) are said to be "critical values" or CVs.

back to top

Redundant Measurements

Increasing the number of redundancies in an adjustment (by taking more measurements) increases the effectiveness of testing. The more measurements there are, the more likely it is that tests will correctly eliminate problems in the adjustment or reject outliers.

Statistically, as the sample size of the measurements approaches the population of all possible measurements (an infinite sample) the results of a precision analysis approaches the "truth". An infinite sample is not possible, but certainly 1000 horizontal angle measurements gives a much better estimate of the precision of each individual angle than 10 measurements. In practice, a measurement scientist gains a knowledge of measurement systems from experience under a variety of circumstances, and can eventually estimate the expected performance for average and unusual conditions from that experience.

back to top

Specific Tests

Estimate of the Variance Factor (Global Test)

The first test which should be applied to any least squares adjustment is the test of the estimate of the variance factor. This test determines whether the residuals of the adjustment are in accord with the precision of measurement obtained from an "infinite sample", also known as the statistical analysis of variance (ANOVA) test. Practically, the test determines whether the residuals are those expected from the precisions of measurement used in the adjustment.

The quantity:
c_r² = v' Q^-1 v
is a test statistic which follows a Chi-squared distribution with r degrees of freedom, where r is the number of redundancies. The expectation of c_r² is the number of redundancies :

The estimate of the variance factor is often said to have a C² distribution, where

The estimate of the variance factor can be tested by specifying a confidence limit, usually 95%, or probability level, a = 0.05, and determining critical values from statistical tables or computation of the probability density function.

The C² distribution is shown below:

A table of critical values is shown below.

Redundancies r	Lower Critical Value a = 0.025	Upper Critical Value a = 0.975
10	0.33	2.05
30	0.56	1.57
60	0.68	1.39
120	0.76	1.27
	1.00	1.00

Examples:

s_o² = 1.75, r = 30, CV = 1.57 \ reject

s_o² = 0.86, r = 120, CV = 0.76 \ accept

If s_o² falls below the lower critical value then either the mathematical model is over-parameterised or the measurement precisions have been under-estimated. If s_o² falls above the upper critical value then either

• the mathematical model is under-parameterised

• the measurement precisions have been over-estimated

• there are gross errors in the measurements.

back to top

Residuals (Local Test)

Once the variance factor test has been carried out, individual residuals can be tested to determine whether they are gross errors. Although such testing may be only absolutely necessary when s_o² fails the global test, it is generally always carried out. Failure of the global test may imply that there are gross errors, but a pass of the global test does not guarantee that there are no gross errors.

If s_o² passes the global test, the quantity n_i = u_i/q_iis the test statistic and has a N(0,1) distribution, where q_i is the weight coefficient of the residual.

If s_o² fails the global test, the Student-t distribution must be used.

The quantity ti =	ui
	soqi

becomes the test statistic and has a T(0, s_o, r) distribution, where r is again the number of redundancies.

Note

1. As r –> X then s_o –> 1, so T(0, 1, X) is equivalent to N(0, 1)

The interpretation of this is that s_o² passing the global test implies that the sample is representative of the population.

2. qi =	r	si is a common approximation.
	n

The Student-t distribution is shown below:

A table of critical values is shown below.

Redundancies r	Lower Critical Value a = 0.025	Upper Critical Value a = 0.975
10	-2.23	2.23
30	-2.04	2.04
60	-2.00	2.00
120	-1.98	1.98
or Normal	-1.96	1.96

Because the distribution is symmetric, the test usually carried out is either

or

depending on whether the global test passes or fails.

Examples:

s_o² passes , CV_N = 1.96, u_i = -3.24, q_i =2.31, | n_i | = 1.40, \ accept

s_o² =1.73 and fails at r = 30, CV_T = 2.04, u_i = 10.35, q_i = 2.50, | t_i | = 3.14, \ reject

Because all residuals in an adjustment are generally correlated, the rejection of measurements must proceed in a step by step fashion, eliminating the largest residuals one at a time. The removal of one measurement in an adjustment may significantly effect the results and change the pattern of errors and the associated test statistics. Hence, the removal of multiple measurements may result in good data being incorrectly discarded.

back to top

Testing Procedure

back to top

Accuracy

• accuracy can only be checked absolutely by comparisons with previously established information

• typically this checking involves computing root mean square errors of differences at check points ( ) , that is survey stations with known coordinates which are deliberately omitted from the fixed stations

• another method is to initially adjust the network using minimal constraints or a free network approach, in either case there are no external influences on the shape of the network

  a 7 (or 4 for a 2D network) parameter transformation is then used to "fit" the free network to all fixed points via post-processing - the residuals from the transformation are then effectively the same as an RMS error of check points

• the problems which are commonly detected by accuracy checking are scale errors and/or reference station coordinate errors

• scale errors are sometimes detected in old surveys because of errors propagated from baselines or from older EDM traverse surveys where calibration or velocity corrections were inaccurate

  the scale error can be modelled by introducing an additional parameter into the network adjustment or by estimation from the post-processing approach

• errors in reference station coordinates (fixed stations used to define the datum of a new survey) are commonly caused because the stations were determined during a previous survey which used older, less accurate equipment

  the errors may be systematic, but in all likelihood will be random within the statistical variations (precisions) of the derived coordinates predicted by the previous adjustment

  this problem is particularly relevant to regional surveys or national geodetic networks covering large areas, which typically realised station coordinate precisions to ±0.1m or poorer, whereas a local, small area survey with modern equipment may be good to ±0.01m

• sequential or phased adjustment (constrained station coordinates) is the answer to this problem, as this technique allows the statistical variations of the "fixed" stations to be accommodated in the adjustment of the new survey - without this sequential adjustment process the new survey will be distorted in shape or scale, and accuracy checks will indicate a poor match to the previous survey

• however sequential adjustment often raises as many problems as it solves

  • previous survey adjustment data/results may be difficult to obtain

  • was the previous survey adequately tested?

  • are the precisions of the "fixed" station coordinates appropriate?

  • are the previous stations original and stable?

  • the new coordinates of the "fixed" stations are commonly ignored

back to top

Reliability

• reliability can be defined as the ability of a survey network to detect errors in the measurements

• reliability is directly related to redundancy, as the more measurements which are available to define the coordinates of a station, then there is a greater possibility of detecting an error in an individual measurement

• to allow a relative gauge of reliability, various types of reliability factors or redundancy numbers are used as indicators

• reliability factors can be determined for each measurement or the network as a whole

factors for each measurement readily show those measurements which have poor reliability

factors for the network as a whole can only be assessed by experience with many such network adjustments

• one common reliability indicator is the Pelzer criterion which is computed by :

  t =	sm	or	t =	sm
  	sv			sm - sl

  where s_m s_l, s_v = the precision of the measurement, adjusted measurement and residual respectively

• the Pelzer factor varies between unity and infinity, the larger the number the poorer reliability

  the factor is effectively based on the variation of the precision of the adjusted measurement, which will be zero in the best circumstances and equal to the precision of the measurement in the worst circumstances

  a practical limit on this factor must be imposed to avoid division by zero, a measurement with no reliability (for example an unchecked radiation) will have the maximum factor

• the global factor is computed as :

  T2 =	1	S (t2 -1)
  	m

  and will vary between zero and infinity (m is the number of measurements)

  T values for networks (not traverses) would normally be in the range of 0.5-2.0

• there are a number of other reliability indicators with different ranges, but all attempt to give a relative measure of the reliability of measurements or networks

back to top