Simple Least Squares A Simple Least Squares Problem: You have surveyed a small levelling network consisting...

Transcribed Image Text:

Simple Least Squares A Simple Least Squares Problem: You have surveyed a small levelling network consisting of five stations (Figure X). Two of the stations in the levelling network are benchmarks with known (fixed) reduced levels. Figure 1: Levelling Network Your observations (six height differences) and the known heights (RLS) of the benchmarks are shown Table X. These measurements are given as the height of the second station above (+) or below (-) the first. Furthermore, the table shows the length of each levelling segment (path length). Fixed RLs Station RL 1 10.000 3 15.000 Observations From To dH (m) d (m) 1 2 1.992 600 2 3 3.018 600 2 4 3.997 300 2 5 5.001 600 3 4 0.997 500 4 5 0.983 400 Table 1: Levelling Data The standard deviation (in metres) of a measured height difference is given as a function of the path length d (in km) as: GAH = 0.015d Tasks: Use the least squares algorithm to estimate: 1. The heights (RLS) of the unknown stations. 2. The standard deviations of the estimated heights (RLS). 3. The adjusted measurements. 4. The standard deviations of the adjusted measurements. 5. The corrections. 6. The standard deviations of the corrections. Finally, identify any correction(s) that fall outside their 90% confidence interval. Simple Least Squares A Simple Least Squares Problem: You have surveyed a small levelling network consisting of five stations (Figure X). Two of the stations in the levelling network are benchmarks with known (fixed) reduced levels. Figure 1: Levelling Network Your observations (six height differences) and the known heights (RLS) of the benchmarks are shown Table X. These measurements are given as the height of the second station above (+) or below (-) the first. Furthermore, the table shows the length of each levelling segment (path length). Fixed RLs Station RL 1 10.000 3 15.000 Observations From To dH (m) d (m) 1 2 1.992 600 2 3 3.018 600 2 4 3.997 300 2 5 5.001 600 3 4 0.997 500 4 5 0.983 400 Table 1: Levelling Data The standard deviation (in metres) of a measured height difference is given as a function of the path length d (in km) as: GAH = 0.015d Tasks: Use the least squares algorithm to estimate: 1. The heights (RLS) of the unknown stations. 2. The standard deviations of the estimated heights (RLS). 3. The adjusted measurements. 4. The standard deviations of the adjusted measurements. 5. The corrections. 6. The standard deviations of the corrections. Finally, identify any correction(s) that fall outside their 90% confidence interval.