8.4 Unit hydrograph theory and applications

Unit Hydrograph Theory

A unit hydrograph captures how a watershed converts rainfall into runoff. It's the direct runoff hydrograph you'd observe if exactly one unit depth of effective rainfall (1 cm or 1 inch) fell uniformly across the entire watershed at a constant rate over a specified duration. Once you have a unit hydrograph for a watershed, you can predict the runoff from any storm by scaling and combining copies of it.

This approach works because of four key assumptions. When those assumptions hold reasonably well, unit hydrographs give you a powerful, practical way to forecast flood peaks and design drainage infrastructure. When they break down, you need to understand why.

Core Assumptions

Linearity. Direct runoff is directly proportional to effective rainfall intensity. If you double the rainfall depth, you double every ordinate of the runoff hydrograph. This is the assumption that makes the whole method work, and it's also the one most likely to fail during extreme events.

Time invariance. The watershed produces the same runoff response for identical rainfall events regardless of when they occur. A 1-hour, 1 cm storm in March produces the same unit hydrograph shape as the same storm in October. In reality, seasonal changes in vegetation and soil moisture violate this to some degree.

Uniformity. Effective rainfall is distributed evenly over the entire watershed area with no spatial variation. For small watersheds this is often reasonable; for large basins where storms may cover only a portion of the area, it becomes a significant simplification.

Superposition. Runoff responses from separate rainfall pulses can be added together. This lets you break a complex, multi-hour storm into individual unit pulses and sum their individual runoff contributions to get the total hydrograph.

Deriving a Unit Hydrograph from Observed Data

Deriving a unit hydrograph requires a well-defined, isolated storm event with relatively uniform rainfall and a clear runoff response. Here's the process:

1. Separate base flow from the observed total runoff hydrograph. Common techniques include the constant discharge method (draw a horizontal line from where the hydrograph begins to rise) or the concave method (which follows the recession curve more closely). The goal is to isolate the direct runoff component.

2. Estimate the effective rainfall hyetograph from observed rainfall data. You need to remove losses due to infiltration, interception, and depression storage. Two common approaches:

   • The phi-index method assumes a constant loss rate ($\phi$) across the storm; any rainfall intensity above $\phi$ becomes effective rainfall.
   • The SCS curve number method uses a curve number based on land use, soil type, and antecedent moisture conditions to estimate total abstractions.

3. Compute the total volume of direct runoff by summing the direct runoff ordinates (after base flow separation) and converting to a depth over the watershed area.

4. Divide each direct runoff ordinate by the total effective rainfall depth. If the storm produced 3 cm of effective rainfall, divide every ordinate by 3. This scales the hydrograph down to represent the response to just 1 cm of effective rainfall.

5. Verify the result by checking that the volume under the unit hydrograph equals 1 cm (or 1 inch) of depth over the watershed area. If it doesn't, normalize by dividing each ordinate by the computed volume so the total equals one unit.

The resulting unit hydrograph has a specific duration tied to the duration of the effective rainfall pulse (e.g., a "1-hour unit hydrograph" or a "6-hour unit hydrograph"). You can only apply it directly to storms discretized into pulses of that same duration.

Unit Hydrograph Applications

Predicting Runoff Using Superposition

Once you have a unit hydrograph, predicting runoff from a new storm is straightforward. Suppose you have a 3-hour unit hydrograph and a storm that produces effective rainfall of 2 cm in hour 1, 5 cm in hour 2, and 1 cm in hour 3.

1. Estimate the effective rainfall hyetograph for the storm event using the phi-index or SCS curve number method.

2. Discretize the effective rainfall into unit pulses matching the unit hydrograph duration. In this example, you'd have three 1-hour pulses of 2 cm, 5 cm, and 1 cm.

3. Scale the unit hydrograph for each pulse by multiplying every ordinate by that pulse's rainfall depth. The first pulse gives you $2 \times UH$, the second gives $5 \times UH$, and the third gives $1 \times UH$.

4. Lag each scaled hydrograph by the appropriate time offset. The first pulse starts at $t = 0$, the second at $t = 1$ hour, and the third at $t = 2$ hours.

5. Sum the lagged hydrographs at each time step. This gives you the total direct runoff hydrograph for the storm.

6. Add base flow to obtain the total runoff hydrograph. Base flow can be estimated from pre-storm conditions or recession analysis.

This process is often organized in a convolution table where each column represents a lagged, scaled unit hydrograph and the row sums give the total direct runoff at each time step.

Limitations

Unit hydrographs are widely used, but you should understand where they fall short:

• Linearity breaks down during extreme events. During very large floods, infiltration capacity may be fully exceeded and other nonlinear processes dominate. The proportional scaling assumption can significantly overestimate or underestimate peak flows.
• Spatial variability is ignored. Real storms don't fall uniformly across a watershed. For large basins (roughly above 5,000 $km^2$), spatial variation in both rainfall and watershed properties like soil type and slope becomes too significant to ignore.
• Results depend heavily on base flow separation and loss estimation. Small errors in separating base flow or choosing a $\phi$-index can propagate into a poorly shaped unit hydrograph. If your input data are uncertain, your unit hydrograph will be too.
• Storage effects cause problems. Watersheds containing large lakes, reservoirs, or extensive wetlands attenuate and delay runoff in ways the unit hydrograph framework doesn't capture well. The linear response assumption doesn't account for the nonlinear behavior of these storage elements.
• Antecedent moisture conditions aren't built in. A dry watershed and a saturated watershed will produce very different runoff responses to the same rainfall, but a single unit hydrograph treats them identically. The SCS method partially addresses this through antecedent moisture condition classes, but only at the loss estimation stage.
• Watershed characteristics change over time. Urbanization increases impervious surfaces and speeds up runoff. Deforestation, agricultural changes, and even climate shifts in precipitation patterns can all invalidate a unit hydrograph derived from historical data. You should periodically re-derive unit hydrographs for watersheds undergoing significant change.

Despite these limitations, unit hydrographs remain one of the most practical tools in applied hydrology. They strike a useful balance between simplicity and predictive power, especially for small to medium watersheds with reasonably uniform characteristics and moderate storm events.