Table Of Contents

• A Few Words On Precision And Accuracy
• Networks
• Redundancy & Reliability
• Statistical Testing
• Survey Network Uses
• Least Squares
• Network Design
• The Shape Of The Earth
• Vertical And Horizontal Observations And Networks
• References
  

A Few Words On Precision And Accuracy

In this subject, and others that deal with least squares and other aspects of surveying, precision and accuracy will often be mentioned. A distinction between the two terms will be useful: a general rule of thumb is that precision refers to repeatability, whilst accuracy refers to the "closeness to the truth". If I measured the distance from Melbourne to Sydney 25 times, and the spread of my measurements was 5mm, I may conclude that my measurements were precise (repeatable). However if the mean of these precise measurements was 32.231128 kilometres my answer, whilst being precise, would not be very accurate.

Consider Figure 1 - here the target shooter has been fairly precise (repeatably hitting roughly the same spot), but assuming the shooter was aiming for the bull's eye they have not been very accurate.
In Figure 2 the shooter has been less precise (the spread is larger), but has been more accurate - that is closer to the bull's eye, than in figure 1.
Figure 3 represents precise and accurate shooting.

Note that quite often in surveying accuracy, or a closeness to the truth, is hard to quantify. This issue will become especially apparent with datums.

back to top

Networks

What is a (good) network - where as many connections between stations as possible are measured, with the goal of introducing redundancy. For our purposes a traverse will not count as a network. The following diagram represents a survey network with good redundancy:

Using geodetic networks (networks covering the continent providing first order control) as an example the evolution of survey networks can be followed. Before EDM most geodetic surveys used triangulation. In triangulation only angles are measured, with very accurate baselines providing scale, and Laplace stations azimuth:

In such networks scale and orientation errors accumulate with distance away from the baseline(s) and Laplace station(s). In these type of networks plan and height coordinates were treated as separate problems.

The invention of EDM allowed the length of all lines in the network to be easily measured, thus alleviating the scale error problems. Most of Australia's geodetic network was surveyed with this technique. Plan and height were still treated as independent problems.

More recently GPS baselines added to such networks also improved scale as well as orientation (the GPS baseline is basically a 3D vector observation). GPS also has the advantage of not requiring a line of sight. Today, with GPS and 3D traversing, most survey networks integrate both plan and height.

With each new measurement technology the achievable measurement precision has increased. This presents a problem for modern surveys where the level of precision obtained can sometimes be several orders of magnitude better than the geodetic frame work.

The aim of virtually all surveys is to produce coordinates - the use of a well designed network (as opposed to a traverse or radiations for example) provides several advantages:

back to top

Redundancy & Reliability

Reliability is the ability of the network to detect errors in the observations. Generally, the greater the redundancy (ie. number of observations) the greater the network's reliability. As networks increase in "importance", for example deformation monitoring, good reliability becomes crucial.

back to top

Statistical Testing

Adjusting a network (with least squares) provides a means for statistically testing the observations to detect gross errors, provides an estimate of the precision of the network's coordinates, and allows the reliability of the network and individual measurements to be estimated.

back to top

Survey Network Uses

• Geodesy - 1^st order control. The geodetic network provides coordination for virtually all other surveys especially the cadastral framework and mapping.
• Control surveys - engineering works, roads, subdivision
• Deformation - dams, bridges, production components. Often in cases of deformation monitoring, failure to detect the deformation can be costly (in human life), for example dam failure

In general a network can be used to solve any surveying problem, and tend to be used where reliability is important.

back to top

Least Squares

When a survey is over-determined - that is there are more measurements than the minimum required to compute the coordinates - the least squares algorithm is commonly used to compute the coordinates, making use of the redundant observations. The least squares algorithm was proposed independently by both Gauss and Legendre in 1795, with Legendre being the first to actually publish its description. Some features of least squares that make it useful for determining the coordinates in a survey network include:

• Least squares allows all observations to be combined using observation weights (and correlations between observations, especially horizontal angles, phased adjustments and GPS baseline components).
• The least squares estimate is an unbiased estimate "…on average the least squares solution is equal to the true solution" (Cross, 1983, p. 98).
• the least squares parameter estimates will be the maximum likelihood estimates.

As mentioned previously least squares also provides an estimate of the parameter (coordinates in this case) precision, and allows for statistical testing of the observations for error detection.

In this course, only the least squares case of observation equations will be used.

back to top

Network Design

The design of a network can often be as much an art as a science. Generally a required coordinate precision must be reached given certain restrictions such as: measurement time, available instrumentation and physical restrictions. Often knowing what can be achieved under such conditions is a matter of experience.

Usually the geometric configuration of a network is designed, and a simulation using certain measurements and precisions is performed to estimate the achievable coordinate precisions and network reliability. If the initial design is not suitable a new configuration of geometry/observations and precisions is tried until a suitable configuration is found. (This process can be carried out in TDVC and is part of the major prac assignment.)

Achieving a certain network criteria can be broken into two parts: instrumentation and field techniques (the measuring of the network) and the network design. The precision and accuracy of a network can be influenced by such things as:

• Instrument choice - use a 20" theodolite or a 1" theodolite, a chain or EDM, a builder's level or precise level? Selecting the appropriate instruments for the task.
• Instrument calibration - ignorance of a prism constant may give a suitable precision, but poor accuracy.
• Field techniques - specific observation time, reciprocal observations, repeating measurements, forced centring.
• Modelling external factors - atmospheric effects, earth curvature.

Issues concerning network design include (most of these issues are merely mentioned here, and will be taken up in detail over the next few weeks):

• Control - how much and whereabouts in the network, is the control likely to distort the network and can I tell if it does.
• Physical constraints - lines of sight, access.
• Intersection geometry - angle intersections, distance intersections, both.
• Propagation of variance.
• Network simulation - knowing the rough coordinates of network stations, measurements and precision of measurements between stations can be varied, and the network simulated to predict the eventual coordinate precision.
• Placement of survey points - static and dynamic considerations.

back to top

The Shape Of The Earth

All survey network adjustments are based on some model of the shape of the earth:

• Plane: used where the ellipsoidal nature of the earth can be ignored. The area over which a flat earth assumption remain valid largely depends the on the required precision of the network. Observation equations are based on plane trigonometry.

• Map grid: network adjustment on the map grid takes some account of the curved earth. It is a step between a planar system and the rigorous solutions (which follow). A map grid approach remains valid for networks up to approximately 20km by 20km. One problem with the map grid approach is surveys that cross zone boundaries. Observation equations are based on plane trigonometry with corrections derived from the map projection.
• Ellipsoid: considers the shape of the earth. The adjustment is performed on an ellipsoid that is a mathematical approximation of the shape of the earth. Observations are usually reduced to the ellipsoid (eg. a slope distance is reduced to an ellipsoidal distance). Observation equations are based on spherical trigonometry.

• Ellipsoid centred cartesian coordinated: again considers the shape of the earth. This model has come into favour recently since it is suited to the use of GPS observations, and GPS baselines are easily implemented as observation equations. Observation equations are based on vector geometry.

We will concentrate on planar network adjustment since it provides the least complicated introduction to survey network adjustment, however all principles remain valid, and are easily extended into the other models listed above.

back to top

Vertical And Horizontal Observations And Networks

Traditionally vertical and horizontal networks were treated separately, especially in adjustments performed on an ellipsoid datum. One major reason for this separation was lack of computational power and the independence of vertical and horizontal observations. Older networks tended to consist of horizontal angles and distances reduced to the ellipsoid to define plan coordinates, and levelling to define height. This approach also reduced the effects of unknown geoid/ellipsoid separation.

Today networks tend to be treated as a 3D problem for several reasons:

• Computation power permits larger adjustments
• Adjustments based on ellipsoid centred cartesian coordinate systems
• 3D measurement techniques such as 3D traversing and especially GPS make separating surveys into plan and vertical components difficult
• Better geoid models (again GPS)

back to top

References

Cross, P. A., 1983. Advances in Least Squares Applied to Position Fixing. Department of Land Surveying Working Paper No. 6, North East London Polytechnic, Essex, England.

back to top