Firstly, connect each station with every other station and with all external points of the catchment's boundary. Let's call the rainfall stations P, Q, R, and S.
The rainfall stations P, Q, R and S are interconnected and also connected with points A, B, C, D and E. Subsequently, the Thiessen polygons will be formed, and the areas of the polygons around each station need to be calculated.
Let's call station P's area as Area P, station Q's area as Area Q, station R's area as Area R and station S's area as Area S.
Given the areas/zone of influence of each station as follows:
Area P = 1200 km^2
Area Q = 1000 km^2
Area R = 1500 km^2
Area S = 1300 km^2
Now, sum up the areas of all the stations, you will get:
Total Area = Area P + Area Q + Area R + Area S = 5000 km^2
This is the total area of the catchment.
To calculate the average rainfall over the entire catchment, each rainfall station's data is weighted by the proportion of the total area that its corresponding Thiessen polygon encompasses. Hence, the average rainfall over the catchment can be calculated by the formula:
Weighted Rainfall = (Rainfall P * Area P + Rainfall Q * Area Q + Rainfall R * Area R + Rainfall S * Area S ) / Total Area
Substituting the given values in the formula:
Weighted Rainfall = (88 mm * 1200 km^2 + 102 mm * 1000 km^2 + 112 mm * 1500 km^2 + 116 mm * 1300 km^2 ) / 5000 km^2
Upon performing this calculation, we get the weighted rainfall to be 105.28 mm.
So, the average rainfall over the catchment using Thiessen polygon method is around 105.28 mm. This method assumes that the rainfall at each station is representative of the rainfall on the entire polygon around the station.