Hydrographs - Theory & Concepts

Hydrographs

Learning Objectives

Define what a hydrograph is and identify its primary components.
Understand the need for and common methods of baseflow separation.
Explain the theory and key assumptions behind the Unit Hydrograph (UH).
Derive a Direct Runoff Hydrograph (DRH) from a UH for both simple and complex storms.
Utilize the S-Curve method to convert a UH to different durations.
Understand the concept and derivation of the Instantaneous Unit Hydrograph (IUH).
Apply Synthetic Unit Hydrograph methods like Snyder's and the SCS Dimensionless UH for ungauged catchments.

Analysis of streamflow over time, breaking down the components of a hydrograph, and utilizing Unit Hydrograph theory to predict watershed response to rainfall events.

What is a Hydrograph?

Hydrograph

A plot of discharge ( $Q$ ) versus time ( $t$ ) at a specific section of a river or channel. It represents the integrated response of a catchment to rainfall inputs.

Components of a Single-Peaked Hydrograph

Rising Limb

The ascending portion of the hydrograph, influenced by the storm character (intensity, duration) and catchment state (wetness).

Crest Segment (Peak)

The highest point ( $Q_p$ ), representing the maximum flow rate.

Recession Limb

The descending portion, representing the withdrawal of water from storage (surface, channel, and ground). Its shape is largely independent of the storm and depends on catchment characteristics.

Lag Time

The time difference between the center of mass of rainfall excess and the peak of the hydrograph.

Baseflow Separation

To analyze the Direct Runoff Hydrograph (DRH)—which results solely from the storm event—the Baseflow (groundwater contribution) must be subtracted from the total streamflow hydrograph.

DRH Calculation

Calculates the Direct Runoff Hydrograph by isolating the direct storm response from the background baseflow.

Q_{DRH} = Q_{Total} - Q_{Baseflow}

Variables

Symbol	Description	Unit
$Q_{DRH}$	Direct Runoff Hydrograph ordinate	$m^3/s$
$Q_{Total}$	Total measured hydrograph ordinate	$m^3/s$
$Q_{Baseflow}$	Baseflow ordinate	$m^3/s$

Common Separation Methods

Separation Methods

Straight Line Method: Connects the start of the rising limb directly to a point on the recession limb.
Fixed Base Method: Assumes baseflow recession continues until the time of peak flow, then rises sharply to meet the recession limb.
Variable Slope Method: Adjusts baseflow dynamically based on historical master recession curves.

Interactive Simulation: Baseflow Separation

Use the simulation below to explore how different baseflow separation methods affect the resulting Direct Runoff Hydrograph.

Unit Hydrograph Theory

Sherman (1932) introduced the Unit Hydrograph (UH), a powerful tool for predicting flood hydrographs from a known storm.

Unit Hydrograph (UH)

The hydrograph of direct runoff resulting from 1 unit (e.g., 1 cm or 1 inch) of effective rainfall occurring uniformly over the basin at a uniform rate during a specified duration ( $D$ ).

Key Assumptions (Linear System Theory)

Time Invariance

The DRH for a given effective rainfall is always the same, regardless of when it occurs (assuming initial conditions are similar).

Linear Response (Proportionality)

Runoff ordinates are directly proportional to rainfall excess volume. For example, 2 cm of rain produces a DRH with ordinates twice that of the 1 cm UH.

Superposition

Hydrographs from consecutive rainfall bursts can be added together (lagged by their respective start times) to produce a composite hydrograph.

Interactive Simulation: Hydrograph Convolution

Experiment with superposition by adding multiple rainfall bursts and observing the composite Direct Runoff Hydrograph below.

Hydrograph Convolution (Superposition)

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Pulse 1 (t=0)1 cm

Pulse 2 (t=1)0 cm

Pulse 3 (t=2)0 cm

This simulation demonstrates the Principle of Superposition. The total hydrograph is the sum of the individual hydrographs generated by each rainfall pulse, lagged by their respective start times.

Interactive Simulation: Unit Hydrograph

Explore the principles of proportionality by scaling the effective rainfall and observing the changes to the Unit Hydrograph below.

Deriving a Unit Hydrograph from a Storm

To derive a UH from an observed storm, engineers must work backward from the total hydrograph.

Unit Hydrograph Derivation Steps

STEP-BY-STEP

Baseflow Separation: Isolate the Direct Runoff Hydrograph (DRH) from the total measured streamflow.
Calculate DRH Volume: Integrate the area under the DRH to find the total volume of direct runoff.
Calculate Effective Rainfall Depth: Divide the DRH volume by the catchment area to find the depth of effective rainfall ( $P_{eff}$ ) in cm or inches.
Determine Storm Duration: Analyze the corresponding hyetograph to determine the uniform duration ( $D$ ) of the effective rainfall burst.
Compute UH Ordinates: Divide every ordinate of the DRH by the effective rainfall depth ( $P_{eff}$ ). The resulting ordinates form the $D$ -hour Unit Hydrograph.

Deriving DRH from UH

Conversely, if we already have a $D$ -hour Unit Hydrograph ( $U$ ) and a storm of excess rainfall $P$ cm (duration $D$ ), the resulting DRH ordinates are:

DRH from UH Calculation

Calculates the Direct Runoff Hydrograph by scaling the Unit Hydrograph by the effective rainfall.

Q_{DRH}(t) = P \cdot U(t)

Variables

Symbol	Description	Unit
$Q_{DRH}(t)$	Ordinate of the Direct Runoff Hydrograph at time t	$m^3/s$
$P$	Effective rainfall depth	cm
$U(t)$	Ordinate of the Unit Hydrograph at time t	$m^3/s/cm$

Deriving DRH from a Complex Storm

A complex storm consists of successive periods of rainfall with varying intensities. Using the principle of superposition, the total DRH is the sum of the individual DRHs produced by each period of effective rainfall, appropriately lagged.

Complex Storm Procedure

If a storm has three successive effective rainfall bursts of duration $D$ : $R_1, R_2, R_3$ . The total DRH ordinate at time $t$ is calculated by scaling the $D$ -hour UH ordinates ( $U$ ) by each burst amount, taking care to lag the time index by the duration $D$ for each successive burst.

Total DRH Ordinate for a Complex Storm

Calculates the total DRH ordinate for a storm with three successive bursts.

Q_{total}(t) = R_1 \cdot U(t) + R_2 \cdot U(t-D) + R_3 \cdot U(t-2D)

Variables

Symbol	Description	Unit
$Q_{total}(t)$	Total Direct Runoff Hydrograph ordinate at time t	$m^3/s$
$R_1, R_2, R_3$	Effective rainfall bursts of duration D	cm
$U(t)$	Ordinate of the D-hour Unit Hydrograph at time t	$m^3/s/cm$
$D$	Uniform duration of each effective rainfall burst	hours

Changing UH Duration: The S-Curve Method

The Unit Hydrograph is derived for a specific storm duration $D$ . The S-Curve Method converts a UH of duration $D$ into a UH of any other duration $T$ (either shorter or longer).

S-Curve Concept

An S-Curve represents the runoff hydrograph resulting from a continuous, infinite sequence of $D$ -hour effective rainfall bursts. It is derived by summing a series of $D$ -hour UHs, each lagged by $D$ hours from the previous one. The curve eventually reaches an equilibrium discharge where the inflow equals the outflow.

Deriving the New Unit Hydrograph

To find a $T$ -hour Unit Hydrograph, shift the original S-Curve by $T$ hours. Subtract the lagged S-Curve from the original S-Curve. This difference represents the runoff from a storm of duration $T$ , but its volume corresponds to $T$ units of rainfall. To normalize the volume back to 1 unit of rainfall, multiply the ordinates by the ratio $(D / T)$ .

S-Curve Conversion Formula

Converts a Unit Hydrograph of duration D to a new duration T using S-Curves.

U_T(t) = \frac{D}{T} [S(t) - S(t-T)]

Variables

Symbol	Description	Unit
$U_T(t)$	Ordinate of the new T-hour Unit Hydrograph at time t	$m^3/s/cm$
$D$	Original duration of the Unit Hydrograph	hours
$T$	Desired new duration	hours
$S(t)$	Ordinate of the original S-Curve at time t	$m^3/s/cm$
$S(t-T)$	Ordinate of the S-Curve lagged by T hours	$m^3/s/cm$

Instantaneous Unit Hydrograph (IUH)

The IUH is a theoretical concept used to describe a catchment's pure impulse response function, completely independent of rainfall duration.

What is an IUH?

The Instantaneous Unit Hydrograph is the direct runoff hydrograph produced by $1 \text{ cm}$ of effective rainfall applied to a catchment instantaneously (duration $D \to 0$ ). Because it removes the dependency on storm duration, it acts as a fundamental characteristic of the basin. Unit hydrographs of any specified duration can be mathematically derived from the IUH using convolution or the S-curve technique.

Clark's Unit Hydrograph Method

Clark's method generates an IUH by modeling the catchment as a combination of pure translation (movement of water) and pure attenuation (storage effects). It first uses a Time-Area Histogram to translate effective rainfall to the catchment outlet based on travel times. Then, it routes this translated hydrograph through a theoretical linear reservoir at the outlet, defined by a storage coefficient $R$ , to account for the catchment's natural attenuation.

Synthetic Unit Hydrographs

For catchments where no streamflow records exist (ungauged catchments), a Unit Hydrograph cannot be derived from a storm. Instead, it must be synthesized using empirical equations relating UH parameters to basin physical characteristics (area, length, slope).

Snyder's Synthetic Unit Hydrograph

Snyder's Method

F.F. Snyder (1938) developed relations between physical characteristics of a drainage basin and the main parameters of its unit hydrograph: time to peak ( $t_p$ ), peak discharge ( $Q_p$ ), and base time ( $t_b$ ). It is heavily reliant on regional coefficients.

Snyder's Lag Time Equation

Calculates the lag time for a synthetic unit hydrograph.

t_p = C_t (L \cdot L_c)^{0.3}

Variables

Symbol	Description	Unit
$t_p$	Basin lag time	hours
$L$	Length of main stream from outlet to divide	km
$L_c$	Length of main stream from outlet to a point opposite the centroid of the basin	km
$C_t$	Regional constant depending on basin topography	-

Snyder's Peak Discharge Equation

Calculates the peak discharge for a synthetic unit hydrograph.

Q_p = \frac{2.78 \cdot C_p \cdot A}{t_p}

Variables

Symbol	Description	Unit
$Q_p$	Peak discharge	$m^3/s$
$C_p$	Regional constant depending on basin storage	-
$A$	Catchment Area	$km^2$
$t_p$	Basin lag time	hours

SCS Dimensionless Unit Hydrograph

Developed by the US Soil Conservation Service (SCS, now NRCS), this method provides a standard dimensionless shape for a unit hydrograph, developed from averaging many UHs from different geographical locations.

SCS Dimensionless Shape

The dimensionless UH is plotted as $Q/Q_p$ on the y-axis against $t/t_p$ on the x-axis. It is defined by coordinates (e.g., peak at $t/t_p = 1$ , $Q/Q_p = 1$ ; terminating at roughly $t/t_p = 5$ ). To generate a specific UH for a basin, one only needs to calculate the time to peak ( $t_p$ ) and peak discharge ( $Q_p$ ) based on catchment area and time of concentration, then scale the dimensionless coordinates to physical units.

Key Takeaways

A Hydrograph is a continuous plot of stream discharge against time, visually representing how a catchment responds to a specific rainfall event.
A typical storm hydrograph consists of a Rising Limb, a Crest Segment (Peak Flow), and a Recession Limb.
Baseflow Separation is required to isolate the Direct Runoff Hydrograph (DRH) for storm response analysis.
The Unit Hydrograph (UH) is the pulse response function of a linear hydrologic system to one unit of effective rainfall over a specific duration $D$ .
UH theory assumes the catchment acts as a linear, time-invariant system obeying principles of Proportionality and Superposition.
Deriving a DRH from a Unit Hydrograph involves multiplying the UH ordinates by the total effective rainfall depth.
The S-Curve method converts a $D$ -hour Unit Hydrograph into a $T$ -hour UH by subtracting lagged S-curves and scaling by $D/T$ .
The Instantaneous Unit Hydrograph (IUH) models the theoretical basin response to an infinitesimally brief burst of rainfall ( $D \to 0$ ), capturing the basin's fundamental routing characteristics.
Synthetic Unit Hydrographs (like Snyder's Method and the SCS Dimensionless UH) allow engineers to estimate runoff for ungauged catchments using empirical regional constants and physical basin dimensions.

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