The Using Scale Factors - Assignment Example

QUESTION 2(150 marks) Write short notes, and prepare suitable diagrams where appropriate, to explain the following: (c) What scale factors are involved in converting coordinates on a UTM map grid to local ground-based coordinates. (40 marks) MGA Point Scale Factor (USQ) = 0.9997371584 Point Scale Factor, k, is the ratio of an infinitesimal distance at a point on the grid to the corresponding distance on the spheroid: k= dL/ds= dS/ds It is the distinguishing feature of conformal projections, such as a Transverse Mercator, that this ratio is independent of the azimuth of the infinitesimal distance. The central scale factor is changed to account for the MGA point scale factor at the centre of the project site and also the datum correction (to raise the map projection to average project height). The point scale factor can be obtained from spreadsheets or manual calculation using Redfearn’s formula. Datum Scale Factor = R/(R+h) = 6365900 / (6365900+760.629) = 0.99988052937 The datum scale factor is simply a ratio of the geometric mean radius of curvature (average of radius of curvature in the meridian and prime vertical planes) of the ellipsoid at the project centre (R) divided by this radius plus the ellipsoid height at that point (R+h). Combined Scale Factor = MGA Point Scale Factor × Datum Scale Factor = 0.99961771917 The combined scale factor is the MGA point scale factor multiplied by the datum scale factor. The MGA projection central scale factor now needs to be changed to a value equal to MGA central scale factor (0.9996) divided by the combined scale factor. Central Scale Factor, k0 = 0.9996 / Combined Scale Factor = 0.99998227405 It is a scale factor on the central meridian. It is superimposed on all projected distances. The central scale factor is changed to account for the MGA point scale factor at the centre of the project site and also the datum correction (to raise the map projection to average project height). (d) Two critical conditions that have to be met when using local ground-based coordinates due to assumptions made in their design. (Hint: without meeting these conditions, grid distances on the local ground-based coordinate system may not accurately represent ground distances.) (30 marks) To ensure conformation to the internal consistency requirements for a particular geometric accuracy class, the following conditions should be met: - The relative error ellipses should confirm the capability of the network design to meet the specifications - The standardised residuals and the variance factor should confirm that the observations have actually met the required standard. QUESTION 3 (150 marks) Write short notes, and prepare suitable diagrams where appropriate, to demonstrate your understanding of the following with respect to network control using static GPS surveying: (a) Minimally and fully constrained network adjustments and how these relate to class and local uncertainty as defined in SP1 (Special Publication 1 - ICSM publication). (50 marks) Minimally constrained adjustment This adjustment is normally performed only after bad baselines have been eliminated using the loop closure programs. Minimally constrained least squares adjustment (often called free adjustment) is a major quality control procedure in a network to be computed. The network design is important in obtaining useful information from an adjustment. A single loop would provide the minimum required information while a network such as the ones in figures below would provide a high degree of redundancy needed for network analysis Loop closures in a GPS network When performing a ‘minimally constrained’ adjustment, the datum coordinates of one point are normally entered, and the coordinates of all vectors are shifted. To agree with the chosen coordinates of the one point. Therefore, a listing of the amounts each point was shifted to be consistent with the fixed point is available and can be printed and analysed. A minimally constrained adjustment allows confirmation of the internal consistency of a network based on the precision assigned to each observation. It also provides a mechanism to detect gross errors and to scale the baseline statistics to get a more realistic error estimate for the baselines. Fully constrained adjustment The fully constrained adjustment transforms the network of observations to the control points in the network. Once the network is fixed to those control points, adjusted coordinates based on the project datum for all other points in the network can then be determined. A fully constrained adjustment is undertaken to fit the survey into the existing coordinate set. Where the observations to be constrained are on a different datum from the one required, they are transformed as part of the adjustment process. For example, in Australia GRS80 ellipsoid associated with GDA might used. In these cases the ease with which the survey can be fitted to the existing coordinate set will be a function of the precision of the transformation technique as well as the precision of the observations and the homogeneity of the existing coordinate set. It is essential that all of these factors are taken into account when performing the constrained adjustment. (b) How Single, Double and Triple Differencing can be used to resolve ambiguities and develop a solution for a GPS baseline. Your answer should explain the general concepts behind the solution of the ambiguities, but does not need to explain the mathematics involved. (50 marks) Single difference between satellites A single difference (SD) can be created by differencing the carrier cycle propagation path lengths from a single satellite (SV1) to two users, (U1 and U2). This is done by subtracting the two equations above. Between receiver single difference Graphically the single difference looks like this (the direction of SV1 has been swapped over to the other side in this diagram): Single difference between receivers Similarly, for the same receivers and a new satellite (SV2) it is possible to structure another ‘single difference’ of a pair ofequations (and at the same epoch in time or at time t zero).Refer to diagram below Receiver clocks are not as accurate as satellite clocks – therefore errors are even larger. GPS receivers measure to all satellites in view. We can remove satellite clock errors by taking one measurement away from another. Single difference between satellites ● Receiver clocks have large errors. ● GPS is a multi-satellite system that permits differencing across satellites. The double difference Both receiver and satellite clock errors can be removed in one go – double differencing (difference of two single differences). Removes clock-errors and reduces many other errors. Has problems with cycle slips (another ambiguity). Still don’t know ambiguities. All observations to a particular Satellite from the one receiver will have the same ambiguity (A) (Provided there are no cycle slips). Double difference ● Made from two Single-Differences ● The basic GPS processing observable for standard GPS surveying applications. ● Reduced 4 one-way observations into one double-differenced observable. ● Double-differences are correlated if formed from observations taken simultaneously. ● Ambiguities still to be estimated. The triple difference If we take two double differences away from each other the result is a triple difference. This will remove the ambiguities. (Related to time). Triple differences can be used to detect cycle slips. Triple difference ● Made from two Double-Differences ● Eliminates the ambiguity terms ● Useful for approximate ‘robust’ baseline solutions. Single difference – 2 observations Double difference – 4 observations Triple difference – 8 observations When software is not sure of integer (whole number) value of ambiguities – double difference ambiguity free solution – ambiguities have any value. When ambiguities are resolved – double difference ambiguity fixed solution – ambiguities held fixed to an integer value. Triple difference A triple difference can now be formed by differencing two double differences at different epochs in time: Graphically the triple difference looks like this: (c) How residual plots can be used to identify possible incorrect ambiguity resolution and possible multipath in processed baselines. (50 marks) Good residual plot Although noisy data is probably the most common problem, other issues can be detected by properly analysing the baseline statistics and summary reports. Figure below shows an example of noisy data. The residual plot shown below demonstrates the situation where the baseline processing software has incorrectly calculated the integer ambiguity solution fix. Incorrect ambiguity fix It demonstrates the situation where multipath may be present in the data collected at one of the stations. A residual plot that seems to trend upwards or downwards such as this is almost certainly symptomatic of a poor ambiguity fix and should be investigated and corrected. Corrective action may be to raise the elevation mask, eliminate some satellites, change start and/or finish times, or as a last resort remeasure the observation. QUESTION 4(200 marks) Assume a client has hired you to survey the network of seven (7) points shown below. To give you an idea of scale, the ground distance from T3 to T4 is approximately 10 kilometres. You have obtained horizontal control coordinates for stations T2, T3 and T7. You have obtained vertical control coordinates for stations T1, T4 and T6. There are no site obstructions at any of the stations. You are required to derive precise horizontal and vertical coordinates for all points. This survey is for densification of an existing geodetic survey network and is required to enable small scale GPS and conventional surveys to use these marks as future control stations. You must use static GPS observations for this work. You have access to four (4) GPS receivers and ancillary equipment. Provide details of a mission plan that you would use to connect and coordinate the stations. All stations must have at least two independent occupations and at least two independent baselines. At least two of the stations must have three independent occupations. You should achieve these criteria with an efficient number of sessions whilst still achieving acceptable network geometry. Your mission plan should show all relevant details including: baselines to be accepted in each session; receiver movements between sessions; independent occupations; and independent baselines. (a) Appropriate number of sessions (30 marks) The minimum number n of sessions in a network with s sites and using r receivers is given by n=(s-o)/(r-o), where o denotes the number of overlapping sites between the sessions. Therefore, n= 7/4 = 2 sessions (b) Network geometry and redundancies (30 marks) The relationship between GPS baselines stations forms the network geometry. At any network, the geometry forms small closed figures that are ‘braced’ and that contain sufficient redundancy and checks which will increase the reliability of the control network and make trouble shooting much easier because as well as making it easier to isolate troublesome baselines. Redundancy refers to observations beyond the minimum number required to compute coordinates for the points observed. The network below has no redundancy. (c) Appropriate independent baselines in each session of the mission plan (80 marks) The total number of baselines that is computed as r (r –1)/2, and only r–1 of those baselines are independent. The rest are formed from combinations of phase data used to compute the independent baselines. In this case, there are 4 receivers measuring at one point each during a single session. Total number of possible baselines = 4 (4–1)/2 = 6. These are drawn on the diagram below. Total number of independent baselines = (4–1) = 3. One combination of these is drawn on the diagram above. (d) Efficiency of session plan and receiver movement (30 marks) To ensure efficiency, the following is done before and during the exercise. 1. Roughly locate both points and control on a map showing roads to use in moving the observers around the project. 2. From reconnaissance and mission planning software, determine the best times to observe. 3. For each session, draw the independent baselines chosen to be observed on map. Move through the project until all points have been observed. 4. Observing the rules for time differences, plan the repeated occupations and observations. Consider redundancy requirements. 5. Measure and record antenna height in two different units at the beginning and before the end of each session. 6. Fill out observation sheet each session. 7. Every one moves every session (where practical). (e) Appropriate independent occupations in mission plan (30 marks) If the GPS receiver is to stay on the same point from one session to the next, the GPS antenna is dismantled and reset over the mark at least 100mm different in height from the previous session. These will help in solving some problems with antenna height measurements or setup errors between sessions. Every station should have at least two independent occupations. Read More

QUESTION 3 (150 marks) Write short notes, and prepare suitable diagrams where appropriate, to demonstrate your understanding of the following with respect to network control using static GPS surveying: (a) Minimally and fully constrained network adjustments and how these relate to class and local uncertainty as defined in SP1 (Special Publication 1 - ICSM publication). (50 marks) Minimally constrained adjustment This adjustment is normally performed only after bad baselines have been eliminated using the loop closure programs.

Minimally constrained least squares adjustment (often called free adjustment) is a major quality control procedure in a network to be computed. The network design is important in obtaining useful information from an adjustment. A single loop would provide the minimum required information while a network such as the ones in figures below would provide a high degree of redundancy needed for network analysis Loop closures in a GPS network When performing a ‘minimally constrained’ adjustment, the datum coordinates of one point are normally entered, and the coordinates of all vectors are shifted.

To agree with the chosen coordinates of the one point. Therefore, a listing of the amounts each point was shifted to be consistent with the fixed point is available and can be printed and analysed. A minimally constrained adjustment allows confirmation of the internal consistency of a network based on the precision assigned to each observation. It also provides a mechanism to detect gross errors and to scale the baseline statistics to get a more realistic error estimate for the baselines. Fully constrained adjustment The fully constrained adjustment transforms the network of observations to the control points in the network.

Once the network is fixed to those control points, adjusted coordinates based on the project datum for all other points in the network can then be determined. A fully constrained adjustment is undertaken to fit the survey into the existing coordinate set. Where the observations to be constrained are on a different datum from the one required, they are transformed as part of the adjustment process. For example, in Australia GRS80 ellipsoid associated with GDA might used. In these cases the ease with which the survey can be fitted to the existing coordinate set will be a function of the precision of the transformation technique as well as the precision of the observations and the homogeneity of the existing coordinate set.

It is essential that all of these factors are taken into account when performing the constrained adjustment. (b) How Single, Double and Triple Differencing can be used to resolve ambiguities and develop a solution for a GPS baseline. Your answer should explain the general concepts behind the solution of the ambiguities, but does not need to explain the mathematics involved. (50 marks) Single difference between satellites A single difference (SD) can be created by differencing the carrier cycle propagation path lengths from a single satellite (SV1) to two users, (U1 and U2).

This is done by subtracting the two equations above. Between receiver single difference Graphically the single difference looks like this (the direction of SV1 has been swapped over to the other side in this diagram): Single difference between receivers Similarly, for the same receivers and a new satellite (SV2) it is possible to structure another ‘single difference’ of a pair ofequations (and at the same epoch in time or at time t zero).Refer to diagram below Receiver clocks are not as accurate as satellite clocks – therefore errors are even larger.

GPS receivers measure to all satellites in view. We can remove satellite clock errors by taking one measurement away from another. Single difference between satellites ● Receiver clocks have large errors. ● GPS is a multi-satellite system that permits differencing across satellites. The double difference Both receiver and satellite clock errors can be removed in one go – double differencing (difference of two single differences).

Read More