Statistical or empirical relationship methods 27

Flood peaks can be determined by early empirical methods, basic flow frequency analysis and probability distribution fitting.

Early empirical methods: Empirical methods for estimating floods were first used

during the 19th _{century and were based on delineating homogeneous regions of flood}

response. The approach was based on plotting the basin’s area against observed flood peaks to form an envelope whose upper limit was the expected regional maximum flood peak (Kovács, 1988). However, the value of these approaches is limited and very reliant upon a relatively large number of representative flood observations. Cordery and Pilgrim (2000) recommended that their use be avoided. In South Africa, the Regional Maximum Flood (RMF) method developed by Kovács (1988), in accordance with the Francou and Rodier (1967) approach, is a frequently used empirical method to determine appropriate safety evaluation flood peaks for dams. The mathematical relationship (Francou and Rodier, 1967) is:

∙ Equation 2.1

where Q

peak is peak flow (m

3 _s-1_{), C and n are regional constants, and A is the basin area}

(km2).

 uncertainty regarding the boundaries of homogeneous regions – this shortcoming is a common drawback of all regional approaches in hydrology (Kovács, 1988);  very large and very small catchments often cannot be accounted for by the regional

approach because of high heterogeneity of hydrological features;

 empirical equations of maximum flood peak envelope curves, dating from the period before 1960, lacked physical meaning and their application was restricted to well-defined areas (Kovács, 1988).

Regional flood frequency analysis is often used to enhance the estimation of flooding probabilities at locations that have short data record length relative to their return periods. In such situations, extreme flow information from a number of sites can be used to compensate for an inadequate temporal representation of the extreme flows at a given location. Regional flood frequency analysis can, therefore, be employed at gauged locations, where information from similar sites that have been gauged is used to assist with the characterisation of the extreme flow regime at the ungauged sites (e.g. Midgley et al., 1994, Mazvimavi, 2003; Bergstrom, 2006). The basic tenet in regionalisation is that, if a relationship exists between model parameters and basin properties which holds true for a gauged basin, then peak flow can be achieved in an ungauged basin which has similar physical attributes. The most common basin attributes that have been used include: climate; topography; vegetation; soil properties (e.g. Chiew and Siriwardena, 2005); annual rainfall; areal potential evapotranspiration (e.g. Boughton and Chiew, 2006); basin area and geology. There are various means by which regionalisation of method can be achieved. These methods include statistical methods, parameter mapping and an a priori estimation method.

Statistical method is based on the regression relationships developed between optimised model parameters and several basin attributes for a number of gauged basins. Frequently, bivariate and multivariate linear and non-linear regressions are developed and then transferred to the ungauged basin (Boughton and Chiew, 2006). Parameter mapping consists of fixing model parameters to average values for the region. This might achievable if the whole region exhibited the same hydrological response to rainfall input. The parameter mapping method relies heavily on the premise of hydrologic similarity between the gauged and the ungauged basins and therefore the delimitation of hydrological response units (HRUs), based on chosen group-defining signatures (Nathan and McMahon, 1990). While the way to define HRUs has been to use geographical proximity, it is not always a reliable method of judging of hydrologic homogeneity. A priori estimation methods fix values based on experience or the use of values adopted from the literature for determining the basin characteristics. The Model Parameter Estimation Experiment (MOPEX) investigated the relationships between physical and hydro-meteorological basin attributes and the parameters of a number of selected hydrologic

models (Wagener et al., 2006; Ao et al., 2006). It is important to note that testing regionalisation approaches involves reserving a proportion of the gauged basins to test the regional parameter estimations. This means that the data set used to establish the regionalisation will be reduced in size. This can be a problem in areas with a limited number of gauged basins – such as in southern Africa. Many regionalisation studies have met with limited success (Franks, 2002). The problems that seem to haunt all the studies are equifinality and parameter interdependence. It has not been easy to be sufficiently confident, with most regionalisation methods, that all the necessary and dominant controls of basin behaviour have been captured in the regionalisation process.

Flood frequency analysis: Design flood estimation approaches can be categorised as either

streamflow-based or rainfall-based (Doran and Pilgrim, 1986). Rainfall-based approaches rely on the ability of a model to convert rainfall into streamflow, while a streamflow-based approach may be performed by a frequency analysis of observed flows where the observations are available and adequate in both length and quality (Pilgrim and Cordery, 1993; Bobee and Rasmussen, 1995; Dirceu et al., 2005; Faber, 2010; Smithers, 2012). The main characteristic of streamflow-based approaches is the primary reliance on observed streamflow data for the development of the approach. The approach assumes that a series of independent observations of flood characteristics (peak flow, flood volume) fits an underlying probability distribution. Three available approaches include:

 At-Site Analysis – based on data available at site of interest;

 At-Site/Regional Analysis – based on data at site of interest and from other hydrologically similar sites;

 Regional Analysis – based on data from hydrologically similar sites.

The flood characteristic series may consist of annual maxima (annual series) or independent peak flows/volumes over a specified threshold (partial series). The use of an annual series is suitable for the estimation of infrequent floods; the partial series provides useful estimates for frequent floods (Laurenson, 1987). Common to the above approaches is the choice of a suitable probability distribution to describe the series of peak flows/volumes. The distributions employed can range from a simple line-of-best-fit (drawn by hand) to complex multi-parameter theoretical probability distributions (e.g. Chow et al., 1988 Schulze, 2000).

As shown by Schulze (2000) and Smithers (2012), the question of selecting an appropriate distribution has received considerable attention in literature, with diverging opinions expressed by various authors. Schulze (2000) questioned whether a suitable probability distribution could be selected, given that any chosen distribution will vary, inter alia, with the season, storm type and duration for each sub-basin. There can also be measurement errors and inconsistency,

non-homogeneity and non-stationarity of data – all of which violate the assumptions made when fitting a distribution to the data. The regional frequency approach uses hydrologically and climatologically similar and nearby locations (Schulze, 2000) and uses data from several sites to estimate the frequency distribution of observed data at each site (Hosking and Wallis, 1995, Pegram and Parak, 2006). This approach assumes that the regionalised variable has the same distribution at every site in the selected region and that data from a region can thus be combined to produce a single regional flood, or rainfall, frequency curve that will be applicable anywhere in the region, when used with appropriate site-specific scaling (Gabriele and Arnell, 1991; Hosking and Wallis, 1995).

Probability distribution fitting: These methods relate the historical flood peak records to a

probability of occurrence. Both at-a-site and regional flood frequency analysis require the fitting of a probability distribution to the data (Pilgrim and Cordery, 1993). Smithers et al., 1997 summarised approaches available for estimating the parameters of a selected distribution which include: Method of Moments (MM); Maximum Likelihood Procedure (MLP); Probability Weighted Moments (PWM); L-Moments (LM); Bayesian Inference and non-parametric methods. The use of L-moments to fit distributions has received extensive coverage in recent literature (Pilon, and Adamowski, 1992; Guttman et al., 1993; Gingras and Adamowski, 1994; Karim and Chowdhury, 1995; Seed, 2001; Alexander, 2002b; Görgens, 2007a; 2007b; QFCI, 2011; Smithers, 2012; Opere et al., 2012). L-moments are reported to have less bias when compared with other techniques (Bílková and Mala, 2012; Smithers, 2012).