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Recession Analysis

The recession curve is the specific part of the flood hydrograph after the crest (and the rainfall event) where streamflow diminishes, refer Baseflow.

The slope of the recession curve flattens over time from its initial steepness as the quickflow component passes and baseflow becomes dominant. A recession period lasts until stream flow begins to increase again due to subsequent rainfall. Hence, recession curves are the parts of the hydrograph that are dominated by the release of water from natural storages, typically assumed to be groundwater discharge. Recession segments are selected from the hydrograph and can be individually or collectively analysed to gain an understanding of these discharge processes that make up baseflow. Graphical approaches have traditionally been taken but more recently analysis has focussed on defining an analytical solution or mathematical model that can adequately fit the recession segments.

Each recession segment is often considered as a classic exponential decay function as applied in other fields such as heat flow, diffusion or radioactivity, and expressed as:

(Equation 1)

Mathematical equation

or

Mathematical equation

where Qt is the stream flow at time t, Q0 is the initial stream flow at the start of the recession segment, a is a constant also known as the cut-off frequency (fc) and Tc is the residence time or turnover time of the groundwater system defined as the ratio of storage to flow.

The term e-a in this equation can be replaced by k, called the recession constant or depletion factor, which is commonly used as an indicator of the extent of baseflow (Nathan and McMahon, 1990). The typical ranges of daily recession constants for streamflow components, namely runoff (0-2-0.8), interflow (0.7-0.94) and groundwater flow (0.93-0.995) do overlap (Nathan and McMahon, 1990). However, high recession constants (eg > 0.9) tend to indicate dominance of baseflow in streamflow. Another parameter interpreted from the recession segment is the recession index (K) which is the time (in days) required for baseflow to recede by one log-cycle ie Q0 to 0.1Q0. A similar index called the half-flow period or half-life, which is the time (in days) for flow to halve can also be calculated. For streams with low baseflow inputs the half-life may be in the range of 7-21 days, while discharge from large stable natural storages can result in a half-life exceeding 120 days (Smakhtin, 2001).

The integrated form of the classic recession function of Equation 1 is

(Equation 2)

Mathematical equation

where St is the storage in the reservoir that is discharging into the stream at time t. This relationship is called a linear storage-outflow model and implies that the recession will plot as a straight line on a semi-logarithmic scale. However, semi-logarithmic plots of individual recessions are commonly curved rather than linear. This is because other natural storages (eg bank storage, wetlands, deeper confined aquifers) can also contribute to baseflow, and these have different regimes of water release to the stream than that of the groundwater stored in the shallow aquifer (Sujono et al, 2004). The recession curve is effectively a composite of water discharged into the stream from multiple natural storages. This coincides with the concept that a catchment is a series of interconnected reservoirs (such as rainfall, snow, aquifers, soil, biomass etc), each having distinct characteristics in terms of recharge, storage and discharge (Smakhtin, 2001).

A curved semi-logarithmic plot for recessions means that the storage-outflow relationship is non-linear. For groundwater discharge from a shallow unconfined aquifer, there are three main reasons for this non-linearity (Van de Griend et al, 2002):

  1. A falling watertable continually decreases the effective thickness of the aquifer and decreases the ability to drain. Declining watertables can also be attributed to other processes other than stream discharge, such as evapotranspiration or groundwater extraction;
  2. The hydraulic conductivity tends to decrease with depth. This is attributed to increased compaction with depth in unconsolidated sediments, and decreased fracturing with depth in hard rock formations;
  3. With prolonged drainage, the lower order stream channels can run dry, leaving only the highest order reaches receiving baseflow.

Another complication is that the recession behaviour for a stream can change through time. This is reflected in variations in the shape of the recession segments found in a stream hydrograph. This is due to variability in such factors as the areal distribution of rainfall, residual storage in connected surface water bodies, catchment wetness, saturated aquifer thickness or depth of stream penetration into the aquifer. Baseflows are also influenced by seasonal effects such as variations in rainfall and evapotranspiration. High evapotranspiration rates during warm weather or active growing seasons can significantly reduce the baseflow component, particularly in shallow watertable areas.

Different approaches have been used in recession analysis to address this non-linearity and variability in recession:

  1. (i.) Approximating the semi-logarithmic plot of the recession curve as three straight lines of different slope (Barnes, 1940). The gradients of these three lines are inferred to be the recession constants for the main streamflow components of runoff, interflow and groundwater flow. The plotting of the three lines is difficult because of the gradual nature of the change in curvature in the recession;
  2. (ii.) Plotting flow ratios (Q0/Qt) instead of flow (Qt) on the semi-logarithmic plot (Hino and Hasebe, 1984) to facilitate better interpretation of the recession;
  3. (iii.) Using a double logarithmic plot of streamflow against time (Hewlett and Hibbert, 1963). Any abrupt change in slope is interpreted to mark the transition from quickflow to baseflow;

  4. (iv.) The correlation method where the current flow Q is plotted on a natural scale against the flow (Qt) at some fixed time interval t previously (eg 2 days before) for each of the recession curves evident in the hydrograph (Langbein, 1938). A line enveloping the traces of these multiple recessions is drawn through the origin to derive the master recession curve. By rearranging the exponential decay (Equation 1) the recession constant k can be derived from the slope of this master recession curve and the lag time interval t.

    (Equation 3)

    Mathematical equation
  5. (v.) The matching strip method involves plotting multiple recession curves derived from the hydrograph on the one semi-logarithmic plot in order of increasing minimum discharge (Toebes and Strang, 1964). Each recession curve is superimposed and adjusted horizontally to produce an overlapping sequence. The master recession curve is interpreted as the envelope to this sequence, and the recession constant k derived from its slope (Equation 3);
  6. (vi.) The tabulation method where data from the multiple recession curves are used to derive the master recession curve and average discharges calculated for the period of the hydrographic record (Johnson and Dils, 1956). Recession periods are tabulated and sorted, and mean discharges calculated for each timestep. This is either done computationally (Boughton, 1995) or by an analytical solution (Singh, 1989);
  7. (vii.) The recession ratio method which analyses the ratios of current flow (Q) to the flow (Qt) at some fixed time intervalt previously (eg 2 days before). A cumulative frequency diagram is plotted to estimate indices such as the median recession ratio (REC50) as a substitute for the recession constant, k (Smakhtin, 2001);
  8. (viii.) The parameter averaging method where the recession function (Equation 1) is fitted for each of the recession segments in the hydrograph. The recession constants that are derived are then averaged (James and Thompson, 1970);
  9. (ix.) Wavelet transform analysisis a technique to break down a signal into its components and applied in such fields as image processing and geophysics. The technique can also be used in hydrograph recession analysis in terms of separating out the low frequency signature of the baseflow. Plots of frequency against time called mean-square wavelet maps are used to derive recession constants (Sujono et al, 2004);

  10. (x.) Using different storage-outflow models or combinations of storage-outflow models to obtain a better fit to the recession curve. The classic exponential decay function (Equation 1) represents a linear relationship between storage and outflow. Other equations have been developed to model discharge from different types of natural storages (Table 1). By combining these equations, discharge from the various natural storages can be better accounted for. For example, a simple option is to add a constant (b) to the linear reservoir equation:

    (Equation 4)

    Mathematical equation

    This provides a better fit to recession curves that stabilise to a constant streamflow over time. This constant flow may represent discharge from a large groundwater storage or from ice or snow reserves. A model based on combining two linear storages has also been used to provide a better fit to the recession curves for a small forested catchment (Moore, 1997). These two storages were interpreted to represent different residence times for water in footslope and upslope zones in the catchment;

  11. (xi.) The Meyboom method uses stream hydrograph data over two or more consecutive years (Meyboom, 1961). The baseflow is assumed to be entirely groundwater discharged from the unconfined aquifer. An annual recession is interpreted as the long-term decline during the dry season following the phase of rising streamflow during the wet season. The total potential groundwater discharge (Vtp) to the stream during this complete recession phase is derived as:

    (Equation 5)

    Mathematical equation

    where Q0 is the baseflow at the start of the recession and K is the recession index, the time for baseflow to decline from Q0 to 0.1Q0.;

  12. (i.) The recession-curve-displacement method is based on the upward displacement of the recession curve during the rainfall event (Rorabaugh 1964; Rutledge and Daniel, 1994; Rutledge, 1998)). The method assumes that baseflow is entirely groundwater discharge from an unconfined aquifer of uniform thickness and hydraulic properties, with the stream fully penetrating the aquifer. On the basis of the algorithms developed, the total recharge to the groundwater system during the rainfall event has been shown to be about twice the total potential discharge to the stream at a critical time (Tc) after the hydrographic peak. Hence, the total volume of groundwater recharge due to the rainfall event (R) can be estimated from the stream hydrograph by:

    (Equation 6)

    Mathematical equation

    (1) where Q1 is the baseflow at the critical time (Tc) extrapolated from the pre-event recession curve, Q2 is the baseflow at the critical time (Tc) extrapolated from the post-event recession curve, and K is the recession index (Figure 1).

Figure 1: Procedure for recession curve displacement method (after Rutledge and Daniel 1993)

Figure 1: Procedure for recession curve displacement method (after Rutledge and Daniel 1993)

  1. Estimate the recession index (K) from the stream hydrograph record

  2. Calculate the critical time (Tc ), using the relationship

    Mathematical equation
  3. Locate the time on the hydrograph which is Tc days after the peak, where streamflow recessions will be extrapolated to
  4. Extrapolate the pre-event recession curve to derive Q1
  5. Extrapolate the post-event curve to derive Q2
  6. Calculate total potential groundwater recharge using these parameters
Table 1: Different storage-outflow models used in recession analysis (Moore, 1997; Griffiths and Clausen, 1997; Dewandel et al,2003)
Conceptual Model Storage-Outflow Relation Recession Function Storage Types Source Comments
Linear reservoir Mathematical equation Mathematical equation General storage

Boussinesq (1877)

Maillet (1905)

Linearised Depuit-Boussinesq equation.

Approximation for short time periods

Horton double exponential Mathematical equation General storage Horton (1933) Transformation of linear reservoir model
Mathematical equation Coutagne (1948)
Mathematical equation Karstic aquifers Padilla et al. (1994) Qc is discharge from low-transmissivity components of karst
Channel Bank Storage Mathematical equation Channel banks Cooper and Rorabaugh, (1963) Variant of linear reservoir. Also used to model evapotranspirative losses
Exponential reservoir Mathematical equation Mathematical equation Throughflow in soil hydraulic conductivity assumed to exponentially decrease with depth
Power-law reservoir Mathematical equation Mathematical equation
Mathematical equation
Mathematical equation

Springs and unconfined aquifers (p = -2)

Soil moisture

Hall (1968)

Brutsaert and Nieber (1977)

 Recessions modelled using p ~ 1.67 (Wittenberg 1994)
Depuit-Boussinesq aquifer storage Mathematical equation Shallow unconfined aquifer Boussinesq (1904) Special case of power-law reservoir for Depuit-Boussinesq aquifer model

Depression Storage

Detention Storage

 Mathematical equation

Surface depressions such as lakes and wetlands,

Overland flow

 Griffiths and Clausen (1997) variant of power-law reservoir
Two parallel linear reservoirs Mathematical equation Mathematical equation Independent aquifers Barnes (1939)
Two serial linear reservoirs Mathematical equation
Mathematical equation 		Mathematical equation
Cavern Storage Mathematical equation Underground caverns in karst terraine Griffiths and Clausen (1997)
Hyperbola reservoir Mathematical equation Ice melt, lakes Toebes and Strang (1964)
Constant reservoir Mathematical equation Permanent snow and ice pack, large groundwater storages Constant stream flow over a finite time period

References