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Thiessen polygon mean rainfall, pentagon catchment

Short answer

Perpendicular bisectors x=75 and y=0 divide the pentagon. Using shoelace method, polygon areas sum to 11000 km² and Σ(A_i·P_i)=1,144,218.75. Mean rainfall = 1,144,218.75/11000 = 104.0 mm.

The shape of a catchment is in the form of a pentagon ABCDE. There are 4 rain gauge stations P, Q, R and S inside the catchment. The position co-ordinates in km are: A(0,0), B(50,75), C(100,70), D(150,0), E(75,-50), P(50,25), Q(100,25), R(100,-25) and S(50,-25). If rainfalls recorded at P, Q, R and S are 90, 105, 114 and 120 mm respectively, determine the mean rainfall by Thiessen Polygon method.

Method: Draw perpendicular bisectors between neighbouring stations P(50,25), Q(100,25), R(100,-25), S(50,-25). The catchment is divided by the lines x=75x = 75 (between P-Q and S-R) and y=0y = 0 (between P-S and Q-R). Each polygon is clipped by the catchment boundary A(0,0), B(50,75), C(100,70), D(150,0), E(75,-50).

      B(50,75)  C(100,70)
   A(0,0)  P    |   Q          D(150,0)
   --------+----+---+-----------
           S    |   R
              E(75,-50)
          x=75 divides left/right, y=0 upper/lower

Polygon areas (by coordinates, shoelace formula):

StationPolygon verticesArea (km²)Rain (mm)A×P
P(0,0),(50,75),(75,72.5),(75,0)3718.7590334687.5
Q(75,72.5),(100,70),(150,0),(75,0)3531.25105370781.25
R(150,0),(75,-50),(75,0)1875114213750
S(0,0),(75,0),(75,-50)1875120225000
Total110001144218.75

Total area check: pentagon area =11000= 11000 km².

Pˉ=∑AiPi∑Ai=1 144 218.7511 000=104.02 mm\bar P = \frac{\sum A_i P_i}{\sum A_i} = \frac{1\,144\,218.75}{11\,000} = 104.02\ \text{mm}

Answer: Mean rainfall by Thiessen polygon method = 104.0 mm.

✓ Answer checked by Claude Sonnet 5.5

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