Methods of reduction of levels

METHODS OF REDUCTION OF LEVELS

 

1. METHOD 1 - HEIGHT OF COLLIMATION METHOD or HEIGHT OF INSTRUMENT METHOD

In this method, Height of the Instrument (H.I.) is calculated for each setting of the instrument by adding Back Sight (B.S.) to elevation of Bench Mark (B.M.). Reduced level of Change Point (C.P.) is then calculated by subtracting Fore Sight (F.S.) from H.I.

For the next setting, H.I. is obtained by adding B.S, taken on Change Point (C.P.) to its Reduced Level (R.L.). This process continues till R.L. of the last point (Fore Sight) is obtained by subtracting staff reading from the height of the last setting of the instrument.

ediate points, R.L.s of them are calculated by subtracting Intermediate Sight (I.S.) from Height of Instrument (H.I.) for that setting. Entries are made in the field book.

Rules

Height of Instrument (H.I.) = R.L. of a B.M. + B.S. reading on that B.M.

R.L. of other staff stations = Height of Instrument (H.I.) – I.S./F.S. readings

After completing above calculations, it is checked for correctness by arithmetical check.

Arithmetical Check

The difference between the sum of the Back Sights (B.S.) and the sum of the Fore Sights (F.S.) should be equal to the difference between Reduced Level (R.L.) of starting Bench Mark (B.M.) and Reduced Level (R.L.) of the closing point as determined by levelling.

E B.S. ~ F.S. = Last R.L. ~ First R.L.

Note: This check verifies only the calculation of R.L. and not the correctness of R.L. This also offers a check for Height of Instrument (H.I.) and R.L.s of Back Sight (B.S.) and Fore Sight (F.S.). The arithmetic check does not involve Intermediate Sights (I.S.).

 

PROBLEM 2:

The following staff readings were observed successively with a level, the instrument having been moved after third and seventh readings:

2.350, 1.625, 0.700, 2.900, 1.955, 1.400, 0.900, 0.600, 1.500, 1.150

Enter the above readings in a page of a field level book. Calculate the R.L.s of all the points by Height of Collimation Method. The first reading was taken with a staff held on a Bench Mark of R.L. 100.000 m. Apply arithmetical check.

Solution

The instrument was shifted after third and seventh readings. Hence, these readings will be entered in the Fore Sight (F.S:) column. Also, the fourth and eighth readings will be entered in the Back Sight (B.S.) column.

Further, the first reading will be entered in Back Sight (B.S.) column and the last reading in Fore Sight (F.S.) column. All other readings will be entered in the Intermediate Sight (I.S.) column.

Rules:

In the Height of Instrument Method or Height of Collimation Method,

Height of Instrument (H.I.) = R.L. of B.M. + B.S. on that B.M. R.L. of any point =

Height of Instrument (H.I.) – I.S. /F.S. of that point

Instrument Setup S₁

Height of Instrument (H.I.) at Sy = 100+ 2.350 = 102.350

R.L. of intermediate point 2 = H.I. at Si - I.S. / F.S. at 2 = 102.350 – 1.625 = 100.725

R.L. of intermediate point 3 = H.I. at Sı- I.S. / F.S. at 3 = 102.350 - 0.700 = 101.650

Instrument Setup S₂

Height of Instrument (H.I.) at S₂ = R.L. of point 3+ B.S. = 101.650 + 2.900 = 104.550

R.L. of intermediate point 4 = H.I. at S₂ - I.S. at 4 = 104.550 - 1.955 = 102.955

R.L. of intermediate point 5 = H.I. at S₂ - I.S. at 5 = 104.550 – 1.400 = 103.150

R.L. of intermediate point 6 = H.I. at S₂ - I.S. at 6 = 104.550 - 0.900 = 103.650

Instrument Setup S₃

Height of Instrument (H.I.) at Sz = R.L. of point 6 + B.S. = 103.650 + 0.600 = 104.250 R.L. of intermediate point 7 = H.I. at S3 - I.S. at 7 = 104.250 – 1.500 = 102.750 R.L. of intermediate point 8 = H.I. at S3 - I.S. at 8 = 104.250 - 1.150 = 103.100

Arithmetical Check

∑B.S. - ∑ F.S. = Last R.L. - First R.L.

5.850 – 2.750 = 103.100 - 100.000 = 3.100.  Hence checked.

 

 

2. METHOD 2: RISE AND FALL METHOD

In this method, the difference of level between the two consecutive points is obtained by comparing the staff readings taken from the same setup of the instrument. This difference indicates whether next point is at Rise or Fall than the previous one. If staff reading is greater, the point is at Fall. If staff reading is smaller, the point is at Rise.

R.L. of any point is found by adding or subtracting the respective Rise or Fall values from the R.L. of the previous point. This procedure is repeated until the last point is reached.

Rule

Difference in level between the two consecutive points in Rise and Fall method is: = First staff reading - Second staff reading = ± Rise / Fall

Note: When the second reading is subtracted from the first one, the positive result means the Rise and the negative result means Fall.

R.L. of any point = R.L. of the previous point = Rise / Fall of that point. Use positive sign for Rise and negative sign for Fall.

After completing the above calculations, it is checked for correctness by arithmetical check.

Arithmetical Check

This arithmetical check also provides check only for calculations and not the result.

In this, the differences between the sum of B.S. and sum of F.S., the sum of Rise and the sum of Fall and the R.L. of Last point and R.L. of First point should be equal.

∑ B.S. ~ ∑ F.S. = ∑ Rise ~ Fall = Last R.L. ~ First R.L.

[That is, ∑ B.S - ∑ F.S = ∑ Rise - ∑ Fall = Last R.L. – First R.L. or

∑ F.S. - ∑ B.S = ∑ Fall - ∑ Rise = First R.L. - Last R.L.]

This method provides a complete check on the Intermediate Sight (I.S.) also, since all the sights are covered in the arithmetic check.

 

PROBLEM 3: Solve Problem 2 by Rise and Fall Method.

The following staff readings were observed successively with a level, the instrument having been moved after third and seventh readings.

2.350, 1.625, 0.700, 2.900, 1.955, 1.400, 0.900, 0.600, 1.500, 1.150

Enter the above readings in a page of a field level book. Calculate the R.L.s of all the points by Rise and Fall Method. The first reading was taken with a staff held on a Bench Mark of R.L. 100.000 m. Apply arithmetical check.

Solution

The instrument was shifted after third and seventh readings. Hence, these readings will be entered in the Fore Sight (F.S.) column. Also, the fourth and eighth readings will be entered in the Back Sight (B.S.) column.

Further, the first reading will be entered in the Back Sight (B.S.) column and the last reading in the Fore Sight (F.S.) column. All other readings will be entered in the Intermediate Sight (I.S.) column.

Calculation of Rise and Fall of Points

Compare B.S. of 1 and F.S. of 2 (Reading on 1 is greater, therefore Rise from 1 - 2)

Difference in levels between staff stations 1 -2 = 2.350 – 1.625 = 0.725 (Rise)

Difference in levels between staff stations 2-3 = 1.625 - 0.700 = 0.925 (Rise)

Difference in levels between staff stations 3-4 = 2.900 - 1.955 = 0.945 (Rise)

Difference in levels between staff stations 4 - 5 = 1.955 – 1.400 = 0.555 (Rise)

Difference in levels between staff stations 5 - 6 = 1:400 - 0.900 = 0.500 (Rise)

Difference in levels between staff stations 6 - 7 = 0.600 – 1.500 = -0.900 (Fall)

Difference in levels between staff stations 7-8 = 1.500 - 1.150 = 0.350 (Rise)

Calculation of R.L. of Points

R.L. of a point = [R.L. of preceding point + Rise] or [R.L. of preceding point - Fall]

R.L. of station 1 is B.M. = 100.000

R.L. of station 2 = 100.000 + 0.725 = 100.725

R.L. of station 3 = 100.725 + 0.925 = 101.650

R.L. of station 4 = 101.650 + 0.945 = 102.595

R.L. of station 5 = 102.595 +0.555 = 103.150

R.L. of station 6 = 103.150 + 0.500 = 103.650

R.L. of station 7 = 103.650 - 0.900 = 102.750

R.L. of station 8 = 102.750 +0.350 = 103.100

Arithmetical Check

∑ B.S. - ∑ F.S. = ∑ Rise - ∑ Fall = Last R.L. - First R.L.

5.850 -2.750 = 4.000 - 0.900 = 103.100 - 100.000

3.100 = 3.100 = .. 3.100

Hence checked.

 

3. EXERCISES

PROBLEM 4: The following staff readings were observed successively with a level:

0.875, 1.235, 2.310, 1.385, 2.930, 3.125, 4.125, 0.120, 1.875, 2.030, 2.765

The first reading was taken with the staff held upon a B.M. of elevation 132.135. The instrument has been moved after the second, fifth and eighth readings. Enter the readings in a field book form. Reduce the R.L. by Height of Collimation Method. Apply the check. (UQ)

PROBLEM 5: The following consecutive readings were taken with a Dumpy Level:

1.25 (B.M. 200.00), 2.00, 2.75, 4.00 C.P., 1.50,

2.50, 0.75, 0.50 C.P., 3.90, 3.20, 1.75, 0.75

Enter the levels in tabular form. Reduce the levels by Rise and Fall Method. Apply usual check.

Hint: Since C.P.s are mentioned after the 4th and 8th readings, those readings (i.e., 4.00 and 0.50) are to be entered in F.S. columns. Readings next to the 4th and 8th readings (i.e., 1.50 and 3.90) are to be entered in B.S. columns.

 

4. COMPARISON OF HEIGHT OF COLLIMATION METHOD WITH RISE AND FALL METHOD