Hydrographs & Unit Hydrograph

The hydrograph

A hydrograph is a graph of discharge against time at a stream section. A storm hydrograph has:

Time parameters:

• Time to peak $t_p$ — from the start of effective rainfall to the peak.
• Basin lag $t_L$ — from the centroid of effective rainfall to the peak (sometimes to the centroid of the hydrograph).
• Time of concentration $t_c$ — time for runoff from the hydraulically most remote point to reach the outlet.
• Time base $T_B$ — duration of direct runoff.

Factors affecting the hydrograph shape

• Catchment shape — fan-shaped catchments: high, sharp peaks; fern-leaf (elongated): flatter, delayed hydrographs.
• Size — large catchments have longer time bases and lower peaks per unit area.
• Slope — steep channels and land slopes give steep rising limbs.
• Drainage density — high density gives quicker, peakier response.
• Land use — forests and vegetation flatten the hydrograph; urban areas sharpen it.
• Storm characteristics — intensity, duration, areal distribution and direction of movement: a storm moving downstream gives a higher peak than one moving upstream.

Baseflow separation

To obtain the direct runoff hydrograph (DRH), baseflow is subtracted:

1. Straight-line method — join the start of the rising limb to a point on the recession limb $N$ days after the peak.
2. Fixed base method — extend the pre-storm recession to below the peak, then join to the point $N$ days after the peak.
3. Variable slope method — separate groundwater recession curves before and after the storm.

Recession curve: $Q_t=Q_0{K}_r^t$, with recession constant $K_r<1$ (separately for surface, interflow and baseflow storage).

Effective rainfall hyetograph

Effective rainfall (rainfall excess) = rainfall − losses (using the φ-index or other loss models). Its volume equals the volume of direct runoff. The ERH plotted against time is the input to unit hydrograph computations.

Unit hydrograph

Assumptions

1. Time invariance — the DRH for a given effective rainfall is always the same regardless of when it occurs.
2. Linear response — ordinates are proportional to the effective rainfall depth (proportionality), and hydrographs from successive storms can be added (superposition).
3. Effective rainfall uniformly distributed over the catchment and within the duration.

Limitations

• Precipitation must be nearly uniform — so the method is best for catchments of moderate size (large catchments are subdivided; the upper limit is often quoted as about 5000 km²).
• Not suitable when snowmelt or large channel storage dominate, or for very small plots.
• The effective rainfall must have a duration close to the unit duration used.

Area under the unit hydrograph

The volume of direct runoff equals 1 cm over the catchment:

Derivation from an isolated storm

1. Select an isolated, uniform storm of duration about D hours.
2. Separate baseflow → DRH.
3. Compute the effective rainfall depth = DRH volume ÷ catchment area.
4. UH ordinate = DRH ordinate ÷ effective rainfall depth (cm).
5. Average the unit hydrographs from several storms (average peak and time to peak; adjust shape to keep unit volume).

Using the unit hydrograph

For effective rainfalls of $R_1,R_2,\dots$ cm in successive D-hour blocks, the DRH is obtained by multiplying the UH by each depth, lagging each by D hours, and adding. Baseflow is then added to get the flood hydrograph.

Changing the unit duration

Method of superposition

A nD-hour UH (n integer) = sum of n D-hour UHs lagged successively by D hours, divided by n.

S-curve method

The S-curve method works for any T (larger or smaller than D, not necessarily a multiple).

Synthetic unit hydrographs

For ungauged catchments, UH parameters are related to catchment characteristics:

• Snyder's method: basin lag $t_p=C_t(L{L}_{ca}{)}^{0.3}$ ($L$ = main stream length, $L_{ca}$ = distance along the stream to the point nearest the catchment centroid, in km; $t_p$ in h); peak discharge $Q_p=\frac{2.78C_{p}A}{t_p}$ (m³/s per cm).
• SCS dimensionless unit hydrograph — time to peak $T_p=D/2+t_L$; peak $Q_p=\frac{2.08A}{T_p}$ (m³/s per cm, A in km², $T_p$ in h); time base ≈ 5$T_p$ for the curvilinear form (2.67$T_p$ for the triangular form).
• Regional methods — in India, the Central Water Commission's flood estimation reports give regional synthetic UH relations for hydro-meteorological sub-zones.

Instantaneous unit hydrograph (IUH)

The UH as $D\to 0$ — the response to 1 cm of effective rainfall applied instantaneously. It depends only on catchment characteristics. Conceptual models: Nash's cascade of linear reservoirs, Clark's model (time–area diagram routed through a linear reservoir). A D-hour UH is obtained by routing/averaging the IUH (or from its S-curve).

Worked examples

Frequently tested points

• Hydrograph: rising limb, crest, recession; recession depends only on catchment storage.
• Fan-shaped catchment and downstream-moving storm → higher peak.
• $N=0.83A^{0.2}$ days; recession $Q_t=Q_0{K}_r^t$.
• UH: 1 cm effective rain, uniform, D hours; assumptions of linearity (proportionality, superposition) and time invariance.
• Area under UH = 1 cm × A: $A=0.36\sum Q\Delta t$.
• S-curve equilibrium $Q_s=2.778A/D$; $U_T=(D/T)[S(t)-S(t-T)]$.
• Snyder $t_p=C_t(L{L}_{ca}{)}^{0.3}$; IUH is the UH with zero duration.