What specific mathematical model is used to derive a direct runoff hydrograph from a given effective rainfall hyetograph for a specific watershed?
The specific mathematical model used to derive a direct runoff hydrograph from a given effective rainfall hyetograph for a specific watershed is the Unit Hydrograph (UH) model. This model operates on the principles of linearity and time invariance to transform effective rainfall into direct runoff.
To understand this, first, a Direct Runoff Hydrograph (DRH) is a graph showing the rate of flow (discharge) of water over time that moves quickly through surface or shallow subsurface pathways to a stream channel, excluding any baseflow (groundwater contribution). An Effective Rainfall Hyetograph (ERH) is a bar chart representing the distribution of rainfall intensity over time that actually contributes to direct runoff, meaning rainfall that has not been lost to infiltration, evaporation, or initial abstractions like surface wetting or depression storage. A watershed is an area of land that drains all the streams and rainfall to a common outlet, such as a river, lake, or ocean.
The core component of this model is the Unit Hydrograph. A Unit Hydrograph is defined as the direct runoff hydrograph resulting from one unit (e.g., 1 centimeter or 1 inch) of effective rainfall uniformly distributed over the entire watershed and occurring at a constant rate over a specified duration (e.g., 1 hour). It essentially represents the watershed's characteristic response, or 'fingerprint,' to a standardized pulse of effective rainfall. Each watershed has a unique Unit Hydrograph for a given duration of effective rainfall.
The Unit Hydrograph model relies on two fundamental assumptions:
1. Principle of Linearity: This principle has two parts. First, proportionality assumes that if the effective rainfall depth changes by a factor, the ordinates (flow values at specific time points) of the resulting direct runoff hydrograph will change by the same factor. For example, if 2 units of effective rainfall occur, the peak flow of the direct runoff hydrograph will be twice that of the Unit Hydrograph. Second, superposition states that the total direct runoff hydrograph from multiple consecutive pulses of effective rainfall is the sum of the individual direct runoff hydrographs produced by each pulse, appropriately shifted in time. Each individual hydrograph is derived by applying the proportionality principle to the Unit Hydrograph.
2. Principle of Time Invariance: This principle assumes that the watershed's response to a given amount and duration of effective rainfall is constant regardless of when that rainfall occurs. In simpler terms, the shape of the Unit Hydrograph remains the same whether the effective rainfall occurs at the beginning or end of a storm.
The mathematical process used to apply the Unit Hydrograph to an effective rainfall hyetograph is convolution, which is essentially a summation process based on the principles of linearity and time invariance. To derive the direct runoff hydrograph, the effective rainfall hyetograph is divided into discrete periods (e.g., one-hour intervals), each with a specific effective rainfall depth. For each interval, this effective rainfall depth is multiplied by the ordinates (flow values) of the Unit Hydrograph, resulting in a partial direct runoff hydrograph. Each subsequent partial hydrograph is then lagged (shifted in time) by the duration of the effective rainfall interval. Finally, all these time-shifted and scaled partial direct runoff hydrographs are summed vertically at each time point to obtain the total Direct Runoff Hydrograph for the entire storm event. This process effectively 'convolves' the watershed's response function (the Unit Hydrograph) with the effective rainfall input (the effective rainfall hyetograph) to predict the runoff output.