In hydrology, estimating the average precipitation over a catchment or a drainage basin is crucial for design and analysis tasks such as flood estimation, reservoir operation, and water resource planning. Since rainfall is recorded at discrete stations, we need techniques to interpolate or estimate how rainfall varies over the entire area. Two of the most commonly used methods for such are:
• Theissen Polygon Method
• Isohyetal Method
These methods provide spatial averaging of rainfall based on either geometrical approximation (Theissen) or contouring based on observed values (Isohyetal). This chapter explores these two methods in full detail, covering construction, assumptions, advantages, and applications.

The Theissen Polygon Method (also known as the Thiessen Method) is a geometrical approach used to estimate area-weighted average rainfall from point observations at rain gauge stations. It assumes that each station is responsible for rainfall over a specific area, bounded by polygons constructed around the stations.

1. Plot the catchment area on a map and mark the location of all rain gauge stations.
2. Connect adjacent stations with straight lines to form triangles (Triangulation).
3. Draw perpendicular bisectors of each line connecting two adjacent stations.
4. The intersection of perpendicular bisectors creates polygons around each station.
5. Each polygon defines the area of influence of that rain gauge.
6. Measure the area of each polygon using a planimeter or CAD tools.
7. Calculate weighted average rainfall as:
   P_avg = Σ(A_i × P_i) / A_total
   • where:
   – A_i = area of the ith polygon
   – A_total = total catchment area
   – P_i = precipitation at ith station.

• Rainfall is uniformly distributed within each polygon.
• No orographic or topographic influence is considered.
• The catchment area is adequately covered by rain gauges.

Merits:
• Simple and easy to apply.
• Useful for small to medium-sized catchments.
• Accounts for spatial variability better than the arithmetic mean method.

Demerits:
• Ignores elevation and topographical features.
• Accuracy depends on the proper distribution of stations.
• Sharp boundaries are unrealistic in natural rainfall distribution.

The Isohyetal Method is a contouring technique used to estimate areal precipitation. It involves drawing isohyets, which are lines connecting points of equal rainfall depth, based on measurements from different stations. This method is considered more accurate than Theissen’s because it incorporates spatial rainfall gradients and allows interpolation between stations.

1. Plot the catchment boundary and mark all rain gauge stations with recorded rainfall values.
2. Interpolate rainfall between nearby stations and draw isohyets (contour lines of equal rainfall).
3. Divide the catchment into zones between isohyets.
4. Measure the area between each pair of isohyets.
5. Assign the average precipitation to each zone, calculated as the mean of the two bounding isohyets.
6. Compute the area-weighted average rainfall as:
   P_avg = Σ(A_i × P_i) / A_total
   • where:
   – A_i = area between two isohyets
   – P_i = average rainfall between two isohyets.

• Use interpolation between stations to place isohyets accurately.
• Rainfall contours must follow logical patterns (e.g., they should not cross unless justified by topography).
• The denser the station network, the more reliable the isohyets.

Merits:
• Most accurate among the non-instrumental methods for estimating areal rainfall.
• Considers spatial variations and gradients of rainfall.
• Can reflect topographic influences when isohyets are drawn carefully.

Demerits:
• Time-consuming and requires experience.
• Subjective element in drawing isohyets—results may vary with person.
• Needs a dense network of rain gauge stations for better accuracy.

Theissen Polygon Method:
• Basis: Geometric (polygonal division)
• Accuracy: Moderate
• Data Requirement: Moderate (location and rainfall data)
• Computation Time: Low to moderate
• Subjectivity: Low
• Spatial Distribution Considered: Yes (simplified)
• Topography Influence: Not considered

Isohyetal Method:
• Basis: Contour (lines of equal rainfall)
• Accuracy: High
• Data Requirement: High (more stations needed)
• Computation Time: High
• Subjectivity: High
• Spatial Distribution Considered: Yes (detailed)
• Topography Influence: Can be considered (qualitatively)

• Theissen Method is commonly used in small watersheds, or where time/resources are limited.
• Isohyetal Method is preferred in hydrologically important or mountainous regions where rainfall gradients are steep and need accurate spatial representation.

Both methods can now be implemented using GIS tools like ArcGIS or QGIS, allowing for:
• Faster area calculation
• Improved accuracy through digital elevation models (DEMs)
• Automated isohyet drawing using spatial interpolation algorithms (e.g., Kriging, IDW)
Such integration reduces subjectivity and enhances the reproducibility of results, particularly for large and complex watersheds.