Problem 1: Thiessen Polygon Method A catchment area has three rainfall stations. Station A covers 40% of the total area and recorded 50 mm of rainfall. Station B covers 30% of the area and recorded 30 mm. Station C covers the remaining 30% and recorded 80 mm. Calculate the average areal rainfall for the catchment.

Problem 2: Isohyetal Method A catchment has isohyets (lines of equal rainfall) drawn at 10 mm, 20 mm, and 30 mm. The area enclosed between the 10 mm and 20 mm isohyets is $40 \text{ km}^2$. The area between the 20 mm and 30 mm isohyets is $60 \text{ km}^2$. Calculate the average areal precipitation over the entire $100 \text{ km}^2$ catchment using the Isohyetal Method.

Problem 3: Optimum Number of Rain Gauges A catchment has 5 existing rain gauge stations. The coefficient of variation ($C_v$) of the rainfall recorded at these stations is calculated to be 30%. If the allowable percentage error ($\epsilon$) in estimating the mean rainfall is set at 10%, determine the optimum number of rain gauges required for the catchment and how many additional gauges need to be installed.

Problem 4: Missing Rainfall Data (Normal Ratio Method) Rain gauge station X was inoperative during a storm. The rainfall recorded at three surrounding index stations A, B, and C during this storm were 45 mm, 52 mm, and 40 mm, respectively. The normal annual average precipitation values for stations X, A, B, and C are $N_X = 800 \text{ mm}$, $N_A = 900 \text{ mm}$, $N_B = 1050 \text{ mm}$, and $N_C = 750 \text{ mm}$. Estimate the missing storm rainfall at station X using the Normal Ratio Method.

Case Study: Orographic Effect on Regional Precipitation The Western Ghats in India and the Sierra Nevada in California are prime examples of the orographic effect heavily influencing regional precipitation patterns. Discuss how this process creates distinct wet and dry zones across a mountain range.

Case Study: Extreme Rainfall and Intensity-Duration-Frequency (IDF) Curves In urban drainage design, engineers rely heavily on historical Intensity-Duration-Frequency (IDF) curves. Discuss how these curves are generated and why they are critical for preventing urban flooding.