Hydrology of Warm Humid Regions (Proceedings of the Yokohama Symposium, July 1993). IAHSPubl. no. 216, 1993. 353

Flood frequency analysis of the Cul de Sac River, St Lucia, using joint probabilities of rainfall and antecedent catchment conditions

M. C. ACREMAN & D. B. BOORMAN Institute of Hydrology, Wallingford, Oxfordshire OX10 8BB, UK

Abstract Flood frequency estimates were prepared for an ungauged site in St Lucia using a novel two-stage process. The first stage was to calculate estimates for a nearby site on the same river for which some data were available; the second stage was to transfer these estimates to the site of interest. During the first stage it was necessary to extend the flood frequency estimates beyond the return-period that could be confidently estimated from the observed data. This was achieved using a rainfall-runoff model that was partly calibrated against observed data, and then used in a joint probability study within a Monte Carlo framework. The technique makes full use of all available data, and has been implemented in a robust fashion that gives considerable confidence in its application.

INTRODUCTION

Estimates of flood frequency are frequently required at ungauged sites throughout the world for the assessment of flood risk and the design of river engineering works. The problem may be tackled by applying a design rainfall to a rainfall-runoff model of the catchment. For the UK, the Flood Studies Report (NERC, 1975) provides methods of defining a set of input variables to a model, including rainfall duration and antecedent catchment wetness, which combine to produce a design flood of required return periods. For other parts of the world where there are fewer data with which to develop such a methodology, hydrologists have employed a rainfall depth of the required return period combined with some "typical" values for other variables such as percentage runoff. However, this does not ensure that the resulting flood will have the same return period as the input rainfall.

This paper concerns an assessment of the flood risk on the Cul de Sac River, St Lucia using a unit hydrograph and losses model. Flood frequency was determined by modelling the joint probability of occurrence of different rainfalls and percentage runoffs using a Monte Carlo simulation approach.

BACKGROUND TO THE STUDY

St Lucia is a 616.4 km2 pear-shaped island forming part of the Lesser Antilles island arc in the eastern Caribbean. The topography of the island is steep and rugged and the flat flood plain of lower Cul de Sac valley offers one of the few sites suitable for

354 M. C. Acreman & D. B. Boorman

Fig. 1 Location map.

urban development (Fig. 1). A 75 ha site was chosen as a development zone containing light industry and shops. The Cul de Sac River, which runs through the development zone, was channelized and flood levees were constructed during a drainage and conservation programme completed in 1987. An initial estimate of the 5-year flood was exceeded on 10 separate occasions during a five month period of the programme implementation, thus a re-evaluation of the flood risk was sought.

RIVER CHANNEL CAPACITY

The first task was to estimate the capacity of the river channel at the development site. Discharge was estimated by the slope/area method using Manning's equation:

Flood frequency analysis of the Cul de Sac River, St Lucia 355

Q = VnSU2R2/3A (1)

where S is the energy slope (in this case the bed slope), R is the hydraulic radius n is the boundary roughness and A is the cross-sectional area.

Parameters A, S and R were determined by surveying of the channel and a value of n was selected from the table provided by Chow (1959) who recommended 0.035 for "an excavated channel with stony bottom and weedy banks". Barnes (1967) provides photographs of 50 stream channels for which roughness coefficients have been determined. Although none is identical to the Cul de Sac, they suggest that a value of 0.035 would not be unreasonable.

A longitudinal profile of the channel bed indicated that the average slope of the reach was 0.0016 which combined with the other parameters yielded channel capacities at two selected sites of around 53 m3 s - 1 and 67 m3 s"1.

The peak discharge during a large event on 15 November 1986 was estimated at 80 m3 s ~ ' (Huntings Technical Services, 1987) and was just retained within the levees. This was made from an estimate of surface water velocity and is likely to be an overestimate since the mean velocity would be somewhat lower. Furthermore, the channel had probably experienced sedimentation since 1986 thus the capacity will have been reduced. Hence the estimates are reasonably compatible.

RIVER FLOW DATA

Since August 1985, the Cul de Sac River has been gauged at Ferrand Bridge some 2 km upstream of the development site. Of the 137 current meter measurements which had been made only five were above 1.3 m, although this water level is exceeded several times per year. Nevertheless, all these data were analysed and an acceptable flood rating curve was derived.

All flood peaks above 2.6 m were extracted from the record and used to produce two series. The first composed the annual maxima floods, the mean of which was 28.8 m3 s_1. The second was the peaks-over-threshold (POT) series of all floods above 23 m3 s_1. From these data the 2-year return period flood (equivalent to a 2.4 year annual maximum flood) was estimated as 30.1 m3 s_1 using the POT model defined by NERC (1975). Floods of higher return period could not be derived with confidence because of a number of gaps in the record and several truncated peaks.

Since the flow data at Ferrand Bridge gauging station were not adequate to define the entire flood frequency curve, it was decided that rainfall data and a rainfall-runoff model would be employed to assess the flood risk.

RAINFALL DATA

Average annual rainfall is around 1300 mm on the coast rising to some 4500 mm in the central highlands. There is no definite wet season, but 60-70% of the rain normally falls between July and November. Rainfall is often very intense for short periods particularly during thunderstorms and cyclones, such as Gilbert (10 September 1985) and Hugo (18 September 1989).

356 M. C. Acreman & D. B. Boorman

Three raingauges were located within the Cul de Sac valley. First, a daily read storage gauge at Soucis near to the development site (see Fig. 1), second an autographic gauge at Bexon School near to the centre of the catchment and third a further autographic gauge at Barre de L'Isle on the watershed. However, the only gauge on the island for which short duration data were extracted on a routine basis was for the Union Agricultural Research Station (Fig. 1). Analysis of daily rainfalls at Union and at Bexon School suggested that the rainfall regimes were not significantly different.

Depth/duration/frequency curves were derived for Union by fitting Extreme Value type I (EV1) statistical distributions to the annual maximum rainfalls of 5, 10, 15, 30, 60, 120, 360 and 720 minutes duration available for the period April 1979 to

140

120

100

80 -

20 -

10 mins.,. ̂

2 L _ l

Return period (years) 10 20 50 ! I

100 _ l

1

Reduced variate

Fig. 2 Rainfall depth/duration/frequency curves.

Flood frequency analysis of the Cul de Sac River, St Lucia 357

September 1987 by the method of probability weighted moments (Fig. 2). Annual maximum rainfalls displayed little variation in depth with return period. This is likely to be due partly to some missing records of extreme rainfalls, but may also result from similar meteorological conditions, which produce intense rainfalls, occurring in most years.

Insufficient data were available to undertake a full analysis of rainfall profiles. Instead, for design purposes symmetrical profiles were constructed for each return period by nesting rainfall depths of various durations using a data interval of one half hour. Catchment average rainfalls were derived by applying an areal reduction factor of 0.9 to the design rainfall profiles.

CONSTRUCTION OF THE RAINFALL-RUNOFF MODEL

A rainfall-runoff model was required to transform design rainfalls over the Cul de Sac valley to design floods. The unit hydrograph and losses model was chosen as the most appropriate model since it is simple and robust with few parameters and has been used widely throughout the world. The loss model separates rainfall into a component that reaches the river quickly, generating flood runoff, termed effective rainfall, and a component which contributes to baseflow and evaporation. Effective rainfall is transformed to response runoff by means of the unit hydrograph.

Ten flood events were initially selected to calibrate the rainfall-runoff model. For each event the rainfalls at the three gauges within the catchment were then examined and four events, which exhibited large spatial variations in rainfall, were discarded, leaving six events for calibration. Details of these are given in Table 1.

Details of the calibration are given in Annex A. The loss model selected was a phi-index, where rainfall less than phi is "lost" whilst that greater than phi is effective rainfall. This is consistent with the concept that the soil has a constant infiltration capacity. For the six events phi ranged from 2.9 mm IT1 to 12.5 mm h - 1 . A strong relationship was found between percentage of the total rainfall lost and river flow before the event (indexing catchment wetness). For flood estimation initial flow was used to define the percentage of rainfall lost and this was distributed through the event using the phi-index model. A unit hydrograph with time to peak of 4 h and peak flow of 0.86 m3 s"1 was derived to transform effective rainfall to flood runoff.

Table 1 Characteristics of the six flood events selected for rainfall-runoff model calibration.

No.

1. 2. 3. 4. 5. 6.

Date

30 March 1989 5 July 1989 24 August 1989 17 September 1989 2 November 1989 31 December 1989

Rainfall:

depth (mm)

109.3 75.6 69.1 49.8 30.8

122.8

duration (h)

23.5 13.0 10.0 7.0 6.0 7.5

Peak flow

(m3 s -1)

25.7 15.3 20.2 25.7 13.5 29.1

358 M. C. Acreman & D. B. Boorman

THE SIMULATION EXERCISE

Since flood runoff rates vary with rainfall and catchment wetness before the storm a similar size of flood may therefore result from heavy rainfall falling on a dry catchment, or more moderate rainfall falling on a wet catchment. The probability of a given flow being equalled or exceeded is therefore a function of the joint probability of occurrence of rainfall and catchment wetness. To model this behaviour a simulation exercise was undertaken in which 10 000 years of synthetic flood data were generated by sampling randomly from the marginal distributions of rainfall and initial flow. Peak flow resulting from each sampled pair of rainfall and initial flow was derived using the rainfall-runoff model. These generated flows were then used to define the flood frequency curve for the catchment.

Full details of the simulation methodology are given in Annex B. The results are shown graphically in Fig. 3 together with the mean annual flood and the 2-year POT flood estimated using the observed flood series. It can be seen that the sets of results are broadly consistent.

It is clear that there is only a small variation in flood peak with return period, for example the 50-year flood is only around 1.6 times larger than the mean annual flood. However, this was somewhat expected since the rainfall frequency curves are also very shallow.

FLOOD FREQUENCY AT THE DEVELOPMENT SITE

In order to calibrate the rainfall-runoff model for the development, the unit hydrograph derived for Ferrand Bridge needed to be adjusted. Packman (1980) lists more than 20 formulae which relate the response time of the catchment (7) to its physical characteristics channel slope (S) and channel length (L) in the form:

T = a (L/S)b (2)

The precise definition of T varies between equations as do the values of a and b, but b is broadly in the range 0.3-0.7.

Estimates of channel length and slope for the catchments to Ferrand Bridge and the development site were derived from the 1:25 000 scale topographical maps and are given in Table 2.

Entering the unit hydrograph time to peak at Ferrand Bridge of 4 h into equation (2) and setting b to 0.5 gives a = 2.24. Replacing the physical characteristics for the development site gives T = 4.7 h (a scaling factor of 1.175). To generate a full unit hydrograph for the development site, all time ordinates for the Ferrand Bridge unit hydrograph were multiplied by 1.175. To retain unit volume dimensionless flow ordinates were multiplied by the inverse 0.851 (1/1.175). These ordinates were made dimensionless by dividing by the Ferrand Bridge catchment area. The unit hydrograph was then re-scaled by multiplying by the drainage area at the development site.

The simulation exercise undertaken to generate the flood frequency curve for Ferrand Bridge was repeated using the rainfall-runoff model calibrated for the development site. The initial flow used for Ferrand Bridge was retained but multiplied

Flood frequency analysis of the Cul de Sac River, St Lucia

80

359

6 0 -

40-

2 0 -

channel capacity

Development site

Ferrand Bridge

POT

Return period (years)

10 20

I I 50 100

_J

0 1 2 3 4 5

Reduced variate y

Fig. 3 Flood frequency estimates for the development (upper curve) and Ferrand Bridge (lower curve and points).

by 1.26, the ratio of the catchment areas (33.64/26.61). Results of flood frequency simulations are shown graphically in Fig. 3. As with the flood frequency curve at Ferrand Bridge, there is only a small variation in flood peak with return period, with the 50-year flood being only around 1.6 times larger than the mean annual flood.

Table 2 Physical characteristics for the Cul de Sac catchment at Ferrand Bridge and at the development site.

Area (km2) L(km) S (m km"1)

Ferrand Bridge

Development site

26.61

33.64

12.4

14.9

15.13

11.32

360 M. C. Acreman & D. B. Boorman

SUMMARY AND CONCLUSIONS

The flow data available for the Cul de Sac River were insufficient to estimate directly floods of high return period for flood risk assessment at a develop site downstream. Instead, simulation experiments were undertaken to determine the joint probability of design rainfalls and given catchment wetness. Unit hydrograph and phi loss models were used to transform rainfall to river flow and results were consistent with low return period estimated from the available flow data.

Surveying of the channel capacity at the development site suggested that flood-plain inundation would occur about once, on average, every 50-60 years. The 5-year flood peak flow was around 54 % higher than the original estimate.

Acknowledgements This study was funded by the Ministry of Planning from the Government of St Lucia and Cul de Sac Industrial Zone Ltd.

REFERENCES

Barnes, H. H. (1967) Roughness characteristics of natural elements. USGS Prof. Pap. 1894. Boorman, D. B. (1990) A new approach to unit hydrograph modelling. PhD thesis, University of Lancaster, UK. Boorman, D. B. & Reed, D. W. (1981) Derivation of a catchment average unit hydrograph. Inst. Hydrol. Report

no. 71, Wallingford, UK. Chow, V. T. (1959) Open Channel Hydraulics. McGraw-Hill, New York. Huntings Technical Services (1987) Drainage and Land Conservation Programme final report. Report to Ministry

of Finance and Planning, Government of St Lucia, November 1987. NERC (1975) Flood Studies Report, 5 volumes. Natural Environment Research Council, London. Packman, J. C. (1980) The effects of urbanisation on flood magnitude and frequency. Inst. Hydrol. Report no. 63,

Wallingford, UK. Tawn, J. A. (1988) An extreme value theory model for dependent observations. J. Hydrol. 101, 227-250.

ANNEX A: CALIBRATING THE RAINFALL RUNOFF MODEL

The unit hydrograph/losses model was chosen as the most appropriate model as it is a simple robust model with few parameters which has been used widely throughout the world. The loss model separates rainfall into a component that reaches the river quickly generating flood runoff, termed effective rainfall and a component which contributes to baseflow and evaporation. Effective rainfall is transformed to response runoff by means of the unit hydrograph.

Two very simple loss models were compared. First the phi-index in which rainfall intensity less than phi is "lost" whilst that greater than phi defines the effective rainfall. This is consistent with the concept that the soil has a constant maximum infiltration capacity. The second loss model considered was the percentage runoff model in which every ordinate contributes the same percentage to effective rainfall, thus the loss rate varies with rainfall intensity reflecting the concept of contributing areas.

Flood frequency analysis of the Cul de Sac River, St Lucia 361

In the first stage of the analysis, the flow hydrograph for each of six events (see Table Al) was separated into response runoff and underlying flow by drawing a straight line from the point of rise of the flood hydrograph to a point on the recession limb a fixed time after the last excess rainfall (this ensures a consistent time base for the event unit hydrographs). This duration was determined by inspection of the flow hydrographs as 18 h. Having identified the volume of response runoff, the percentage and phi-index rainfall separation models were applied to the six events and the catchment average unit hydrograph derived by the method of superposition developed by Boorman & Reed (1981). The two unit hydrographs derived using the alternative loss models were very similar.

Table Al Model parameters of the six flood events.

No.

1. 2. 3. 4. 5. 6.

Rainfall depth (mm)

109.3 75.6 69.1 49.8 30.8

122.8

Response (mm)

56.6 16.1 23.0 28.3 7.8

39.1

flow phi-index (mm

1.46 4.98 3.95 2.20 3.06 6.25

'Ah'1) Percentage (%)

48.2 78.7 66.8 43.2 25.4 31.9

loss Initial flow (m3 s -1)

0.47 0.15 0.09 1.89 0.19 0.29

To determine the appropriate form of loss model the technique of excess rainfall detection proposed by Boorman (1990) was adopted. By deconvolving the average unit hydrograph with the separated rainfall (under a series of appropriate constraints) a series of detected effective rainfalls were calculated. The detected ordinates were the rainfalls that would be needed to best reproduce the response runoff hydrograph when combined with the unit hydrograph. By examining the detected rainfalls, and comparing them with rainfall separated from known separation models, it is was possible to say what form of model should be used. Figure Al shows for one event the observed rainfall (solid line) divided using the phi-index method into excess or effective rainfall (above the line) and rainfall loss (below the line).

From this figure it is immediately clear that a percentage runoff model would be inappropriate to describe the variation of losses within an event. However, the phi-index model is giving a separation quite similar to the detected rainfalls.

From Table Al it can be seen that the values of phi show no strong relationship with the independent variables. However a reasonable relationship exists between the percentage loss and the initial flow (seen plotted in Fig. A2) which shows high losses (when the initial flow is low) that rapidly decrease as initial flow increases. This relationship has been formalized into:

PLOSS = 50 (IFLOW + 0.044)"019 (Al)

This equation was therefore adopted for the model. It is applied by calculating the loss based on the initial flow, and then applying this loss to the total rainfall ordinates using a phi-index.

362 15.0

<D O X

LU

-15.0 - 1

Rainfall

\

Observed

Detected

30.0 —

O

E

5 o

Unit hydrograph

Event

Catchment average

Flood hydrograph

Observed flow

Estimated flow

Baseflow

Recession

0 12 24

Time (hours) Fig. Al Examples of flood event analysis.

36 48

observed modelled

0 . 4 0 . 6 0 . 8 1 1 .2 1 .4

i n i t i a l flow (cumecs)

Fig. A2 Percentage loss plotted against initial flows.

Flood frequency analysis of the Cul de Sac River, St Lucia 363

ANNEX B: THE SIMULATION EXERCISE

It is clear from Annex A that runoff rates vary both with rainfall and catchment wetness as indexed by the initial flow. A similar size of flood may therefore result from heavy rainfall on a dry catchment, or more moderate rain falling on a wet catchment. The probability of a given flow being equalled or exceeded is a function of the joint probability of occurrence of rainfalls and catchment wetness. This could be calculated by numerical integration of the marginal distributions of rainfall and initial flow, though this is a complex mathematical problem. An alternative method is to sample randomly from the marginal distributions of rainfall and initial flow. The river flow resulting from each sampled pair of rainfall and wetness can be derived using the rainfall-runoff model. Then these generated flows can be used to define the flood frequency curve for the catchment.

A significant implication of the interaction described above is that the annual maximum rainfall may coincide with dry antecedent conditions and thus may not generate the annual maximum flood. In contrast, perhaps the second largest rainfall in the year may occur when the catchment is already wet and may produce the year's largest flood. Consequently, for rigorous implementation of this joint probability methodology, all rainfalls and all antecedent wetness through the year should be considered and not just the annual maximum. This would require complete computerized records of 30-min rainfalls and river flows, which were not available.

In lieu of this information, it was considered to be adequate to analyse the three largest rainfalls in a year since one of these would most likely generate the annual maximum flood. Tawn (1988) has shown that the distribution of the second and third largest events can be derived easily from the distribution of the annual maxima in the following way.

The largest rainfall, Ru in a hypothetical year can be derived by generating a random number, Ux, between 0 and 1 from a uniform distribution and then entering this into the inverse distribution function of the annual maximum series:

i?, = -In (-ln(î/i)) a + u (Bl)

which is the inverse of the EV1 distribution where a and u are parameters for the appropriate rainfall duration. The second largest rainfall, R2, in the year is defined by generating a random number, U2, between 0 and Ul from a uniform distribution and then entering this in equation (Bl) using the same values for parameters a and u as for the largest event. The third largest rainfall, R3, in the year is defined by generating a random number, U3, between 0 and U2 from a uniform distribution and so on. This process can be continued to define rainfalls of lower rank order.

Annex A showed that the percentage of rainfall lost during an event was related to the initial flow in the river. Initial flow was derived for 27 flood events from the Ferrand Bridge records. These varied from 0.051 to 3.27 m3 s"1 translating to a range of percentage loss from 68.2% to 26.6% when applied to equation (Al). Figure Bl shows these data presented as a cumulative distribution function represented by the equation:

F(IFLOW) = 0.222 In IFLOW + 0.666 (B2)

364 M. C. Acreman & D. B. Boorman

0 0 .5 1 1.5 2 2 .5 3 3 .5 4 4 .5

initial flow (cumeos)

Fig. Bl Cumulative distribution fonction of initial flows.

whose inverse is:

IFLOW = e(F(IFLOW)-0.666)/0.222 ^ 3 )

This inverse fonction was used to generate random initial flows by replacing F(IFLOW) with a random number between 0 and 1 from a uniform distribution.

In the simulation exercise 10 000 years of synthetic flood records were generated in order to define the flood frequency relationship on the Cul de Sac River at Ferrand Bridge.

For each synthetic year the three highest rainfalls of durations from 30 to 510 minutes were derived using equation (Bl). These were used to define design rainfall to the rainfall-runoff model after applying an areal reduction factor of 0.9.

An independent initial flow was generated randomly for each of the three rainfalls using equation (B3). The value of initial flow was used to define the percentage loss by equation (Al). The catchment rainfall inputs were then separated to produce an effective rainfall using the percentage loss applied as a phi loss rate. Design flows were next derived by transforming the effective rainfall profile to a storm flow hydrograph using the unit hydrograph. Baseflow equal to the initial flow was added to the storm runoff to produce a total runoff hydrograph.

The 10 000 synthetic annual maxima were placed in rank order and floods of various return periods were determined directly from the ranked data using the Gringorten formula:

T = 1/1 - (i - 0.44)/(« + 0.12) (B4)

where n is the total number of annual maxima (in this case 10 000) and i is the z'th maximum (with 1 being the smallest).