Hydrographs - Theory & Concepts

Hydrographs

Analysis of streamflow over time, breaking down the components of a hydrograph, and utilizing Unit Hydrograph theory.

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

\text{DRH Ordinates} = \text{Total Hydrograph Ordinates} - \text{Baseflow}

Common Separation Methods

Separation Methods

The Straight Line Method connects the start of the rising limb to a point on the recession limb. The Fixed Base Method assumes baseflow recession continues until the peak, then rises to meet the recession limb. The Variable Slope Method adjusts baseflow based on master recession curves.

Baseflow Separation Methods

Select a separation method to isolate the Direct Runoff Hydrograph (DRH) from the Baseflow.

Straight Line MethodFixed Base MethodVariable Slope Method

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Straight Line Method:Connects the start of the rising limb directly to the point on the recession limb where direct runoff is assumed to end. It's the simplest method but least physically accurate.

Unit Hydrograph Theory

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

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. (e.g., 2 cm of rain produces a DRH with ordinates 2x that of the UH).

Superposition

Hydrographs from consecutive rainfall bursts can be added (lagged by time) to produce a composite hydrograph.

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.

Unit Hydrograph & Superposition Principle

Demonstrates the principles of Linearity (multiplying the UH by rainfall depth) and Superposition (adding lagged hydrographs).

Effective Rainfall per Storm (cm): 2 cm

Linearity: DRH = UH × Rainfall Depth.

Time Lag for Second Storm (hrs): 0 hrs

Superposition: Total Runoff = Sum of individual storm hydrographs. Set lag > 0 to simulate two consecutive storms.

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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 Calculation

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

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:

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

S-Curve Method

The S-Curve Method converts a Unit Hydrograph of duration

D

into a Unit Hydrograph of any other duration

T

(either shorter or longer). It is derived by summing a series of

D

-hour UHs lagged by

D

hours continuously, representing runoff from an infinite storm.

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, and multiply the ordinates by the ratio

(D / T)

to normalize the volume back to 1 unit of rainfall.

S-Curve Conversion

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

Instantaneous Unit Hydrograph (IUH)

The IUH is a theoretical unit hydrograph resulting from 1 unit of effective rainfall applied to the basin in an infinitely small duration (

D \to 0

). It is purely a function of the catchment properties, independent of storm duration. Modern methods like Clark's IUH use time-area histograms and routing to derive it.

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 (accounting for translation). 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. The resulting hydrograph is the IUH.

Synthetic Unit Hydrographs

For catchments where no streamflow records exist (ungauged catchments), a Unit Hydrograph can be synthesized using empirical equations relating UH parameters to basin 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

Snyder's Equations

t_p = C_t (L \cdot L_c)^{0.3} \\ Q_p = \frac{2.78 \cdot C_p \cdot A}{t_p}

Variables

$t_p$ : Basin lag (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)
$C_t$ , $C_p$ : Regional constants depending on basin topography and storage
$Q_p$ : Peak discharge ( $m^3/s$ )
$A$ : Catchment Area ( $km^2$ )

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.

Instantaneous Unit Hydrograph (IUH)

The IUH is a theoretical concept used to derive Unit Hydrographs for any 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

). It describes the catchment's pure impulse response function, independent of rainfall duration. Unit hydrographs of any specified duration can be mathematically derived from the IUH using convolution or the S-curve technique.

Key Takeaways

A Hydrograph is a continuous plot of stream discharge against time.
It visually represents how a catchment responds to a specific rainfall event over time.
A typical storm hydrograph consists of a Rising Limb, a Crest Segment (Peak Flow), and a Recession Limb.
The shape of the rising limb is heavily influenced by the storm's characteristics, whereas the recession limb depends primarily on the physical properties of the catchment.
To analyze direct storm response, Baseflow Separation is required to isolate the Direct Runoff Hydrograph (DRH).
Methods range from simple straight-line techniques to more complex variable slope analyses based on historical recession data.
The Unit Hydrograph (UH) is the pulse response function of a linear hydrologic system to one unit of effective rainfall over a specific duration.
It assumes the catchment acts as a linear, time-invariant system.
Proportionality dictates that runoff ordinates scale directly with rainfall volume.
Superposition allows adding lagged hydrographs from successive rainfalls to build a complex storm hydrograph.
Deriving a DRH from a Unit Hydrograph involves simple multiplication of 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, eliminating the duration dependency.
Synthetic Unit Hydrographs allow engineers to estimate runoff for ungauged catchments where no historical streamflow data exists.
Snyder's Method utilizes empirical regional constants and physical basin dimensions (length, area) to synthesize the key parameters ( $t_p$ , $Q_p$ ) of the unit hydrograph.
The SCS Dimensionless Unit Hydrograph provides a universal shape that only requires scaling by $t_p$ and $Q_p$ to generate a basin-specific UH.

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