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Answer (1)
For BM1BS = 2.345For TP1BS = 3.015, FS = 2.654For TP2BS = 2.126, FS = 3.678For TP3BS = 3.785, FS = 3.897For BM2FS = 2.982This is about finding corrected elevations using the rise and fall method in a levelling survey. We're surveying from BM1 to BM2 in a straight lineThe known elevations areBM1 = 350.000 mBM2 = 348.040 mThe allowable error is A.E = ±24√k where k is the distance in kilometersFirst setup: distance from BM1 to TP1 is 80 mSecond setup: distance from TP1 to TP2 is 2 × 80 m = 160 mThird setup: distance from TP2 to TP3 is 3 × 80 m = 240 mFourth setup: distance from TP3 to BM2 is 4 × 80 m = 320 mWe have backsight (BS) and foresight (FS) readings in the table.The rise and fall method compares consecutive readings to determine if the ground rises or falls between points.For each pair of consecutive points:If BS > FS, the ground risesIf BS < FS, the ground fallsThe amount of rise or fall = BS - FSFrom BM1 to TP1:BS at TP1 = 3.015FS at BM1 = 2.654Difference = 3.015 - 2.654 = +0.361 (rise)From TP1 to TP2:BS at TP2 = 2.126FS at TP1 = 3.678Difference = 2.126 - 3.678 = -1.552 (fall)From TP2 to TP3:BS at TP3 = 3.785FS at TP2 = 3.897Difference = 3.785 - 3.897 = -0.112 (fall)From TP3 to BM2:BS at BM2 is missingFS at TP3 = 2.982We can't directly calculate this, but we'll find it another wayNow, we calculate elevations. Starting with BM1's known elevation (350.000 m), we'll calculate each point's elevation:TP1 elevation = BM1 elevation + Rise (or - Fall)= 350.000 + 0.361 = 350.361 mTP2 elevation = TP1 elevation + Rise (or - Fall)= 350.361 - 1.552 = 348.809 mTP3 elevation = TP2 elevation + Rise (or - Fall)= 348.809 - 0.112 = 348.697 mBM2 elevation - we know this should be 348.040 mNext, calculate height of instrument (HI) for each setup.HI at BM1 = BM1 elevation + BS = 350.000 + 2.345 = 352.345 mHI at TP1 = TP1 elevation + BS = 350.361 + 3.015 = 353.376 mHI at TP2 = TP2 elevation + BS = 348.809 + 2.126 = 350.935 mHI at TP3 = TP3 elevation + BS = 348.697 + 3.785 = 352.482 mWe verify with closing error.The closing error = Computed BM2 elevation - Known BM2 elevationIf the elevation of BM2 is computed from TP3:BM2 elevation = HI at TP3 - FS to BM2 = 352.482 - 2.982 = 349.500 mBut the known value is 348.040 m, so the error is:349.500 - 348.040 = 1.460 mThis seems quite large, which suggests there might be an issue with the readings or calculations.Next, check if the error is within allowable limits.Total distance = 80 + 160 + 240 + 320 = 800 m = 0.8 kmAllowable error = 24√k = 24√0.8 = 24 × 0.894 = 21.456 mm = 0.021456 mOur error of 1.460 m exceeds this by quite a bit.Lastly, we adjust elevations. Since there appears to be an error in the readings or calculations, we should distribute the error proportionally among all points. However, given the large error value, I recommend double-checking all readings first.Let's verify our calculations with a different approach - using height of instrument method as a check.From BM1: TP1 elevation = HI - FS = 352.345 - 2.654 = 349.691 mFrom TP1: TP2 elevation = HI - FS = 353.376 - 3.678 = 349.698 mFrom TP2: TP3 elevation = HI - FS = 350.935 - 3.897 = 347.038 mFrom TP3: BM2 elevation = HI - FS = 352.482 - 2.982 = 349.500 mThese values are different from our first calculations, which indicates there may be an error in the recorded values.
