Precipitation & Its Measurement

Precipitation & General aspects of Hydrology

1. Index of Wetness

The Index of Wetness gives a simple percentage measure of how wet a particular year is relative to the long-term average at a place.

Formula: Index of wetness = (rainfall in a given year at a given place / average annual rainfall of that place) × 100

• % Rain deficiency = 100 - Index of wetness

Use: It is used for quick annual comparisons, drought monitoring and simple water-resource assessments. A very low index indicates a deficient year and may trigger water-management measures.

2. Aridity index

The Aridity index (A.I.) quantifies the degree to which climate at a location is arid by comparing the water demand of the atmosphere to water actually supplied by evapotranspiration.

Formula: A.I. = ((PET - AET) / PET) × 100

Where:

• PET = Potential evapotranspiration (the evapotranspiration that would occur under unlimited water supply)
• AET = Actual evapotranspiration (the evapotranspiration actually occurring given available moisture)

Classification given (as commonly used in hydrology texts):

• A.I. ≤ 0 → Non-arid
• 1 ≤ A.I. ≤ 25 → Mild arid
• 26 ≤ A.I. ≤ 50 → Moderate arid
• A.I. > 50 → Severe arid

Remarks: A negative or zero A.I. implies available water meets or exceeds atmospheric demand (non-arid). Larger values indicate increasing aridity. PET can be estimated by empirical methods (e.g., Thornthwaite) or energy-balance methods (e.g., Penman-Monteith); PET depends on radiation, temperature, humidity and wind.

3. Optimum number of rain gauges (N)

To estimate how many gauges are required in a basin for a desired precision, the following empirical relation is used:

Formula: N = (C_v / ε)²

Where:

• C_v = coefficient of variation of rainfall = σ / P̄ (σ is standard deviation of station values; P̄ is mean rainfall)
• ε = allowable fractional error (expressed in same units; e.g., for 10% error, ε = 0.10)
• N = required number of independent gauges

Interpretation: Higher spatial variability (larger C_v) or stricter accuracy (smaller ε) increases the required number of gauges.

Notes on computation:

• Compute mean rainfall P̄ from the available station normals.
• Compute σ from station normal values.
• Compute C_v = σ / P̄ and substitute into the formula.

4. Estimation of missing rainfall data

Missing records at a station are commonly estimated from nearby stations. Two widely used approaches are shown below. A practical rule is to use the simpler arithmetic mean when neighbouring stations have similar normals; use the normal-ratio method when station normals differ appreciably.

Arithmetic mean method (simple neighbouring average)

Use when the long-term normals of nearby stations are close to the normal of the station with missing data (difference within about 10%).

Formula: P_x = (P₁ + P₂ + ... + P_m) / m

Where P_x is the estimated rainfall at station x, and P₁, P₂, ... are observed rainfalls at m neighbouring stations. A minimum of three nearby stations is recommended when possible.

Normal-ratio method

Use when neighbouring station normals differ by more than about 10% from the normal of the station with missing data. This method adjusts observations by the ratio of long-term normals and gives greater weight to stations whose normals are closer to the target station.

Formula: P_x = Σ (P_n × N_x / N_n) / Σ (N_x / N_n)

Where:

• P_n = observed rainfall at neighbouring station n for the same period
• N_n = long-term normal rainfall at neighbouring station n
• N_x = long-term normal rainfall at station x (station with missing data)

Remarks: The normal-ratio method preserves differences in climatic normals and reduces bias that would arise if station normals are dissimilar. Always check the reasonableness of estimates (e.g., compare with nearby stations and regional patterns).

5. Mean rainfall over a catchment

Point rainfall from a number of gauges must be converted into an areal average for hydrological and hydraulic computations. Three standard methods are used:

(i) Arithmetic average method

Take the simple arithmetic mean of the station values:

Formula: P̄ = (P₁ + P₂ + ... + P_n) / n

Where P₁...P_n are the rainfall values at the n gauges. This method assumes uniform spatial distribution and is used rarely in practice because it ignores the spatial location of stations.

(ii) Thiessen polygon method

This method assigns weights to each gauge proportional to the area closer to that gauge than to any other. Construct perpendicular bisectors between neighbouring stations to form polygons; the rainfall at each station is assumed to represent rainfall over its polygonal area.

Formula: P̄ = Σ (P_i × A_i) / A_total

Where:

• P_i = rainfall at station i
• A_i = area of the Thiessen polygon around station i
• A_total = total catchment area

Remarks: The Thiessen method is simple, accounts for station location and is superior to the arithmetic mean where station spacing is uneven.

(iii) Isohyetal method

The isohyetal method uses contours of equal rainfall (isohyets). Measured amounts at stations are plotted and isohyets are drawn by interpolation. The areal average is computed by multiplying the mean rainfall in each band (between successive isohyets) by the area of that band, summing for all bands and dividing by the total area.

Procedure (summary):

• Plot station rainfalls on a map and draw isohyets of equal rainfall.
• Identify areas between adjacent isohyets (bands) and determine the area of each band.
• Assign a representative rainfall value to each band (commonly the average of the bounding isohyet values).
• Compute P̄ = Σ (P_band × A_band) / A_total.

Remarks: The isohyetal method is the most accurate when station density is sufficient and rainfall fields are spatially variable, but it is also the most labour-intensive and requires expert interpolation of isohyets.

Summary: Use the index of wetness for quick annual comparisons and the aridity index to quantify dryness. Select the method for estimating missing data according to similarity of station normals (arithmetic mean when similar; normal ratio when not). For areal averages prefer Thiessen polygons or the isohyetal method rather than simple arithmetic mean when station distribution or rainfall variability demands it.