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Estimating future sea level extremes under conditions of sea level rise
Coastal Engineering, 14 (1990) 295-303 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
295
Estimating future sea level extremes under conditions of sea level rise W.E. Bardsley 1, W.M. M i t c h e l l 2 a n d G . W . L e n n o n 2 tDepartment of Earth Sciences, Universityof Waikato, Hamilton (New Zealand) 2School of Earth Sciences, Flinders University, Bedford Park, Adelaide, S.A. 5042 (Australia) (Received October 5, 1988; revised and accepted November 20, 1989)
ABSTRACT Bardsley, W.E., Mitchell, W.M. and Lennon, G.W., 1990. Estimating future sea level extremes under conditions of sea level rise. CoastalEng., 14:295-303. Standard techniques of extreme event analysis involve making the assumption that future annual maxima can be approximated as independent random variables from a common distribution. This approach is not applicable to the investigation of future extreme sea levels, given the possibility of greenhouse-related sea level rise. Instead, design criteria for coastal structures must allow for future maxima to be generated from a sequence of different distributions. Analysis is still quite straightforward in principle, and attention is drawn to some basic statistical results for generating design estimates when the distributions of maxima differ just in the magnitude of the location parameter. An example from Adelaide, South Australia, is used to illustrate the generation and application of predictive models under the multi-distributionalframework. Given sea level rise, quantiles of extreme magnitudes are suggested as being of more practical value in design work than return period magnitudes.
INTRODUCTION
Annual sea level maxima have long been used as the basic input data for extreme value analysis at ports and other coastal locations (Lennon, 1963; Suthons, 1963 ). The annual maxima are either applied directly or undergo an initial correction procedure ifa trend is present (e.g. Walden and Prescott, 1983 ). Some general comments on the use of annual sea level maxima are given by Graft ( 1981 ) and Walden et al. ( 1981 ). Difficulties can arise when translating the results of annual maxima analyses to the problem of estimating future sea level peaks, given some specified scenario of greenhouse-related sea level rise. For example, Blackman and Graft (1978) utilise an equation given by Suthons (1963) for predicting future magnitudes under conditions of a continued upward linear trend in the an0378-3839/90/$03.50
© 1990 Elsevier Science Publishers B.V.
296
W.E. BARDSLEY ET AL.
nual maxima. At the same time, these authors express reservations concerning the validity of the Suthons equation which they recognise as "intuitive". The purpose of this paper is to illustrate how some standard properties of sample maxima can be applied to generate models for the prediction of future extremes given a specified sea level rise. A specific extreme value model is derived by way of an example. The overall approach is stochastic in nature and assumes the availability of a reasonably long sequence of recorded annual maxima. This can be contrasted with more process-related methods such as that of Middleton and T h o m p s o n (1986). These authors were not explicitly concerned with sea level rise but their model could easily be modified to produce an alternative approach to that considered here. PREDICTION FRAMEWORK
The usual assumption is made that the annual sea level maxima at a given locality represent independent random variables. The independence assumption is reasonable since the magnitude of the m a x i m u m in one year is unlikely to influence the m a x i m u m level in the following year. However, the evident presence of trends in maxima at some localities suggests that recorded sequences of annual maxima cannot in general be assumed to be identically distributed (Graff, 1979, 1981 ). In any case, a future period of sea level rise implies that all localities will experience different distributions of maxima from one year to the next. It is reasonable, therefore, to utilise an annualmaxima prediction framework based on sequences of different distributions. Proceeding to the necessary design expressions for a sequence of distributions, define the random variable X, as the m a x i m u m sea level at a given locality with respect to the calendar year t. The associated distribution function is symbolised as F ( x ) , It is assumed that any short-period wave component has been removed from the Xt's (waves of period less than 15 min). Considering first the "return period", define R as the quantity analogous to the usual return period in a stationary sequence of maxima. Measuring time in integer years, the R-year value associated with some specified sea level x is obtained as the integer R giving the nearest solution to: r+R
1-F(x),= 1
(1)
where z is a specified base year from which time is measured. The corresponding expression for continuous time is given by Middleton and Thompson (1986). When all the F(x)t are the same, Eq. 1 reduces to the usual return period expression l / [ 1 - F ( x ) ]. The return period analogue is not very helpful as a practical design aid because the risk factor (probability of non-exceedance) involved with a given
FUTURE SEA LEVEL EXTREMES UNDER CONDITIONS OF SEA LEVEL RISE
297
R is not immediately evident. Instead, it is more useful to use as design criteria the quantile magnitudes, symbolised here as HpT,. In words, H is that specific elevation which has a probability p of non-exceedance by the highest sea level which will occur in the time interval starting with calendar year z and ending in calendar year T. From the standard expression for the distribution function of the largest of a number of different random variables, specified values of z, T, and p determine HpT~as the value o f x obtained from the solution of: T
p=l-[F(x)t /=r
(2)
The base year z might be "this" year for the purposes of producing diagrams showing the current estimate of risk probabilities given a specified rising sea level. For design purposes, z would be some year in the future corresponding to the time of first operation of the coastal structure concerned. The idea of quantile magnitudes can be applied of course to stationary sequences of maxima as well. In this case the start and finish years are of no significance since the quantile magnitudes will be the same for any constant M-year time interval. For large M, the return period magnitude corresponds to the specific quantile magnitude Ho.368, where p = e - 1. The M-year quantile magnitudes of river floods are discussed in NERC ( 1975, p. 237 ). GENERATING PREDICTION EXPRESSIONS
While Eq. 2 is a basic statistical expression, it is not particularly helpful in itself for design work. For example, each annual maximum is deemed to be generated from a different unknown distribution, making distribution selection impossible. However, it is possible to obtain prediction expressions for HpT~magnitudes by making some reasonable assumptions concerning both the form of the F(x)t's and the nature of the changes in these distributions from one year to the next. A simple model for year-to-year change in the F(x)t's is suggested by the usual approach for trend-correcting recorded annual maxima. That is, the observed maxima are adjusted by adding or subtracting various amounts depending on some smoothing function passed through the data (where this "data" might be mean sea level values or the maxima themselves). The adjusted values are then analysed as if they were random variables generated from a common distribution. This procedure makes the implicit assumption that the unadjusted annual maxima arise from distributions which differ only in their degree of translation along the x-axis. In other words, any time trend in the annual maxima reflects the same trend in the location parameters of
the F(x)t's. The assumption of time-stability for all F(x)t parameters except location parameters is utilised in this paper. This seems a reasonable simplification
298
W.E. BARDSLEY ET AL.
provided that differences between individual F(x)t's arise mainly from differences in mean sea level. An implication of this basic assumption is that any trend in future mean sea levels will result in an exactly similar trend in the expected values of the annual maxima. Of course, a future upward trend in mean sea levels implies some form of climate change, which in turn suggests possible changes in the nature and frequency of storm surges. How such changes might alter the shape and scale parameters remains unknown. The approach utilised here therefore must be interpreted as a first-order approximation only. The method for generating prediction expressions can now be summarised. First, for trend-free data, proceed in the usual way by fitting a selected distribution to the recorded annual maxima. This will yield estimates of all unknown parameters. Alternatively, fit the distribution concerned to adjusted annual maxima if trend correction is required (making the assumption that the distributions of annual maxima differ only in location parameter values). When working with maxima which have been trend-adjusted (by whatever means), it is desirable to standardise the adjusted values to some selected year of record. This yields shape and scale parameter estimates, plus a location parameter estimate for the base year concerned. At this point, some checks should be made concerning the goodness of fit of the distribution concerned - using either a graphical method or calculating some formal test statistic. This applies to both trend-free or trend-corrected data. Given a reasonable fit, the next step is to obtain an estimate of the location parameter for year z (taken here to represent "the present" year). For a trendfree record of maxima, the required estimate is just the location parameter estimate for the distribution concerned. For trend-corrected maxima, the initial location parameter estimate by definition is with respect to the year selected for standardisation. This estimate can be converted to that of year z by using the trend previously established (making the assumption that any trend in maxima is a reflection only of a corresponding trend in location parameters). With the location parameter established for year r, any specified linear or non-linear trend in future mean sea levels will automatically define the sequence of location parameters of the F(x),'s to come. That is, if some future year t is specified to have a mean sea level greater by a certain amount d than year z, then the location parameter for year t is just that of z with d added. The mechanism of specification of future mean sea levels is not the concern of this paper. The mean levels may be derived from a deterministic climatic model, or possibly they may simply represent a specified "worst case" scenario for design purposes. At this stage, all the parameters are now either specified or estimated from data. These parameter values are now inserted into Eq. 2 to yield the required H,,T, estimates. This approach is quite general and need not even be confined
FUTURE SEA LEVEL EXTREMES UNDER CONDITIONS OF SEA LEVEL RISE
299
to parametric specification of the F(x)t's. For example, the parameter-free method of Bardsley ( 1988 ) could also be used here. EXTREME VALUE DISTRIBUTIONS
It is well known that any number of distributions can describe a given data set equally well, so we do not advocate any particular distribution or estimation procedure with respect to the F (x)t's. However, because the extreme value distributions have a long history in coastal design work, we give here their respective Newton's method calculation equations for obtaining the required x values in Eq. 2. The three extreme value distributions can be defined:
F(x)t=exp{-exp[- (x-~It)/O]}
Type 1
(3)
-b
where 0, ~ and a are (positive valued) scale parameters and c and b are (positive valued) shape parameters. The quantities q, to and ~ are location parameters, shown here with a t subscript to indicate their time-varying nature in the present context. The Newton iteration expressions for the Type 2 and Type 3 cases are given, respectively, in Eqs. 6 and 7 below as:
xj+l=xj~
~-blnp+ ~,f=r(Xj--tot)-b bE 7=~(xj_ to,)_(b+ 1)
(6)
a q n p + Y~L~(~t --xj) C Xj+l =xj 4
cy~L~(~t_xj)C_,
(7)
For the Type 1 (Gumbel) distribution, an explicit solution to Eq. 2 can be obtained:
HpT~=O-ln(-lnp)+ln[,~exp,
/a~
(8)
EXAMPLE
As an illustration of the kinds of results to emerge from the prediction method, an extreme value analysis was applied to a sequence of 60 annual sea level maxima obtained over the record period 1928-1987 at Port Adelaide
300
W.E. BARDSLEY ET AL.
(Outer Harbor), South Australia. A regression analysis of monthly means of recorded sea levels from this site indicated a linear increase in mean sea level of 2.85 m m per year. The original data were "raw" in that no attempt was made to isolate the meteorological component of the sea level values. The water level values were obtained from a stilling well, which d a m p e d out the influence of waves less than a few minutes in period. The 2.85 m m per year trend was assumed to apply to the maxima which were then adjusted by standardising with respect to the year 1928. We adopted this approach rather than directly testing the maxima themselves for a trend since such small changes would be very difficult to detect amongst the inevitable 'noise' of the extreme sea levels. However, such inferences from means to extremes do need to be treated with some caution (Walden and Prescott, 1983). As seen in Fig. 1, the trend-adjusted maxima on a Gumbel plot indicate an underlying curve with a slope decreasing with increasing y, where y = - l n [ - l n F ( x ) ] , and a numerical value for F(x) was estimated for each data point using the Weibull plotting position expression. This form of plot suggests the possibility of a Type 3 extreme value distribution (henceforth abbreviated to EV3 ). The suitability of the EV3 distribution for describing the data is supported by the reasonable fit of an EV3 prediction function:
x=~-aexp(-c-ly)
(9)
The particular prediction function shown as the curved line in Fig. 1 was obtained by three-point graphical fitting (Bardsley, 1989 ), yielding the parameter estimates of ~, a and c as 2.42 m, 0.71 m and 3.79, respectively. Although a reasonable data fit is achieved, we make the point again that we are not advocating any particular estimation procedure or distribution for 2"3
,
,
,
I
I
-0-75
0
0'75
1"50
2"25
,
,
c~2.1 I
t~ !
-1.50
3-00
3"75
Z+.50
H Fig. I. E x t r e m e v a l u e p l o t o f t r e n d - a d j u s t e d a n n u a l sea level m a x i m a , w i t h f i t t e d T y p e 3 e×t r e m e v a l u e p r e d i c t i o n f u n c t i o n . Site is O u t e r H a r b o r , P o r t A d e l a i d e ( A H D = A u s t r a l i a n H e i g h t D a t u m ).
301
FUTURE SEA LEVELEXTREMESUNDERCONDITIONS OF SEA LEVELRISE
general application. No doubt other distributions and estimation methods could be found which produce marginal improvements of fit to this data set. The location parameter value of 2.42 m is with respect to the standardised year 1928. On the basis of an annual increase in ~t of 2.85 m m per year since 1928, ~199o is obtained as 2.60 m. With z= 1990, subsequent ~t values were calculated on the basis of the intermediate future sea level rise given by Scenario II (NRC, 1987, p. 29). The values ofr~, ~ and ~, were then inserted into Eq. 7 to obtain H0.5T1990 through to T = 2100. The resulting curve is shown in Fig. 2, together with the upper bound line and the Scenario II curve for mean sea level at this location. More conservative design quantile curves such as H0.95 T1990 could also be added as required. The Ho.5T199o c u r v e in Fig. 2 shows an initial rapid increase reflecting the increase in the EV3 quantiles with increasing sample size. However, the rate of convergence to the limiting upper bound value is very slow and the curves come to be dominated by the specified sea level rise as time goes on. For design purposes, the upper bounds are best interpreted from a pragmatic viewpoint as a magnitude with an unspecified low probability of exceedance. Any upper bound value can only be an estimate of the true magnitude. Also, it would no doubt be possible to conceive of some combination of maximum surge and astronomical tide which would exceed the estimated bound for a given year. The upper bound is of course a consequence of the selection of an EV3 distribution, where the random variable cannot exceed ~t. For the EV3 or any other distribution possessing an upper bound, it is im-
ill . . . . . . . 3.0
<
up er bo~n6
2-0-
• e~T~ S
~
~
9redid e~ 1'5
1990
I
I
2020
I
I
I
I
I
2050 20 0 YEAR Fig. 2. Estimated upper bound and Ho.sT~99ofor sea level maxima at Port Adelaide, on the basis o f translated EV3 distributions. The mean sea level for the intermediate scenario is shown in the lower curve, raised by 1.7 m for graphing purposes.
302
W.E.BARDSLEYET AL.
portant to make the distinction between the HpT~and upper bound magnitudes with respect to time applications. The upper bound is with respect to the year concerned - that is, the highest water level in year T should not exceed the bound. On the other hand, the HpT~values state that there is a probability p of non-exceedance by the single highest sea level event between z and year T, without regard as to when that event occurs. CONCLUSION
The standard statistical methodology discussed here should prove useful as a simple means of obtaining practical design estimates under conditions of a rising sea level. If required, extension to the "true" maxima is achieved by simply redefining the annual maxima to be with respect to instantaneous water level. The design procedure should not be interpreted as a scientific forecast since too little is known at present about the factors influencing the future behaviour of mean sea levels. Rather, this technique is a means by which some selected scenario of mean sea level rise can be translated through to a scale of extreme event risk. The "factor of safety" in this case comes in at the start of the analysis with the selection of a rate of sea level rise which errs on the side of caution. ACKNOWLEDGEMENTS
For making available the sea level maxima used here, we are grateful to the Department of Marine and Harbours and the Coastal Branch of the Department of Environment and Planning (South Australia). REFERENCES Bardsley, W.E., 1988. Toward a general procedure for the analysis of extreme random events in the earth sciences. Math. Geol., 20:513-528. Bardsley, W.E., 1989. Graphical estimation of extreme value prediction functions. J. Hydrol., 110: 315-321. Blackman, D.L. and Graff, J., 1978. The analysis of annual extreme sea levels at certain ports in southern England. Proc. Inst. Civ. Eng., 65, Part 2: 339-357. Graff, J., 1979. Concerning the recurrence of abnormal sea levels. Coastal Eng., 2 (2): 177-187. Graff, J., 1981. An investigation of the frequency distributions of annual sea level maxima at ports around Great Britain. Estuarine Coastal Shelf Sci., 12: 389-449. Lennon, G.W., 1963. A frequency investigation of abnormally high tidal levels at certain west coast ports. Proc. Inst. Civ. Eng., 25: 443-449. Middleton, J.F. and Thompson, K.R., 1986. Return periods of extreme sea levels from short records. J. Geophys. Res., 91: (C10): 11,707-11,716. NERC (U.K.), 1975. Flood Studies Report, Vol. 1. Whitefriars Press, London. NRC (US), 1987. Responding to Changes in Sea Level (Engineering Implications). Natl. Academy Press, Washington, D.C.
FUTURE SEA LEVEL EXTREMES UNDER CONDITIONS OF SEA LEVEL RISE
303
Suthons, C.T., 1963. Frequency of occurrence of abnormally high sea levels on the east and south coast of England. Proc. Inst. civ. Eng., 25: 443-449. Walden, A.T. and Prescott, P., 1983. Identification of trends in annual maximum sea levels using robust locally weighted regression. Estuarine Coastal Shelf Sci., 16:17-26. Walden, A.T., Prescott, P. and Weber, N.B., 1981. Some important considerations in the analysis of annual maximum sea levels. Coastal Eng., 4(4): 335-342.
295
Estimating future sea level extremes under conditions of sea level rise W.E. Bardsley 1, W.M. M i t c h e l l 2 a n d G . W . L e n n o n 2 tDepartment of Earth Sciences, Universityof Waikato, Hamilton (New Zealand) 2School of Earth Sciences, Flinders University, Bedford Park, Adelaide, S.A. 5042 (Australia) (Received October 5, 1988; revised and accepted November 20, 1989)
ABSTRACT Bardsley, W.E., Mitchell, W.M. and Lennon, G.W., 1990. Estimating future sea level extremes under conditions of sea level rise. CoastalEng., 14:295-303. Standard techniques of extreme event analysis involve making the assumption that future annual maxima can be approximated as independent random variables from a common distribution. This approach is not applicable to the investigation of future extreme sea levels, given the possibility of greenhouse-related sea level rise. Instead, design criteria for coastal structures must allow for future maxima to be generated from a sequence of different distributions. Analysis is still quite straightforward in principle, and attention is drawn to some basic statistical results for generating design estimates when the distributions of maxima differ just in the magnitude of the location parameter. An example from Adelaide, South Australia, is used to illustrate the generation and application of predictive models under the multi-distributionalframework. Given sea level rise, quantiles of extreme magnitudes are suggested as being of more practical value in design work than return period magnitudes.
INTRODUCTION
Annual sea level maxima have long been used as the basic input data for extreme value analysis at ports and other coastal locations (Lennon, 1963; Suthons, 1963 ). The annual maxima are either applied directly or undergo an initial correction procedure ifa trend is present (e.g. Walden and Prescott, 1983 ). Some general comments on the use of annual sea level maxima are given by Graft ( 1981 ) and Walden et al. ( 1981 ). Difficulties can arise when translating the results of annual maxima analyses to the problem of estimating future sea level peaks, given some specified scenario of greenhouse-related sea level rise. For example, Blackman and Graft (1978) utilise an equation given by Suthons (1963) for predicting future magnitudes under conditions of a continued upward linear trend in the an0378-3839/90/$03.50
© 1990 Elsevier Science Publishers B.V.
296
W.E. BARDSLEY ET AL.
nual maxima. At the same time, these authors express reservations concerning the validity of the Suthons equation which they recognise as "intuitive". The purpose of this paper is to illustrate how some standard properties of sample maxima can be applied to generate models for the prediction of future extremes given a specified sea level rise. A specific extreme value model is derived by way of an example. The overall approach is stochastic in nature and assumes the availability of a reasonably long sequence of recorded annual maxima. This can be contrasted with more process-related methods such as that of Middleton and T h o m p s o n (1986). These authors were not explicitly concerned with sea level rise but their model could easily be modified to produce an alternative approach to that considered here. PREDICTION FRAMEWORK
The usual assumption is made that the annual sea level maxima at a given locality represent independent random variables. The independence assumption is reasonable since the magnitude of the m a x i m u m in one year is unlikely to influence the m a x i m u m level in the following year. However, the evident presence of trends in maxima at some localities suggests that recorded sequences of annual maxima cannot in general be assumed to be identically distributed (Graff, 1979, 1981 ). In any case, a future period of sea level rise implies that all localities will experience different distributions of maxima from one year to the next. It is reasonable, therefore, to utilise an annualmaxima prediction framework based on sequences of different distributions. Proceeding to the necessary design expressions for a sequence of distributions, define the random variable X, as the m a x i m u m sea level at a given locality with respect to the calendar year t. The associated distribution function is symbolised as F ( x ) , It is assumed that any short-period wave component has been removed from the Xt's (waves of period less than 15 min). Considering first the "return period", define R as the quantity analogous to the usual return period in a stationary sequence of maxima. Measuring time in integer years, the R-year value associated with some specified sea level x is obtained as the integer R giving the nearest solution to: r+R
1-F(x),= 1
(1)
where z is a specified base year from which time is measured. The corresponding expression for continuous time is given by Middleton and Thompson (1986). When all the F(x)t are the same, Eq. 1 reduces to the usual return period expression l / [ 1 - F ( x ) ]. The return period analogue is not very helpful as a practical design aid because the risk factor (probability of non-exceedance) involved with a given
FUTURE SEA LEVEL EXTREMES UNDER CONDITIONS OF SEA LEVEL RISE
297
R is not immediately evident. Instead, it is more useful to use as design criteria the quantile magnitudes, symbolised here as HpT,. In words, H is that specific elevation which has a probability p of non-exceedance by the highest sea level which will occur in the time interval starting with calendar year z and ending in calendar year T. From the standard expression for the distribution function of the largest of a number of different random variables, specified values of z, T, and p determine HpT~as the value o f x obtained from the solution of: T
p=l-[F(x)t /=r
(2)
The base year z might be "this" year for the purposes of producing diagrams showing the current estimate of risk probabilities given a specified rising sea level. For design purposes, z would be some year in the future corresponding to the time of first operation of the coastal structure concerned. The idea of quantile magnitudes can be applied of course to stationary sequences of maxima as well. In this case the start and finish years are of no significance since the quantile magnitudes will be the same for any constant M-year time interval. For large M, the return period magnitude corresponds to the specific quantile magnitude Ho.368, where p = e - 1. The M-year quantile magnitudes of river floods are discussed in NERC ( 1975, p. 237 ). GENERATING PREDICTION EXPRESSIONS
While Eq. 2 is a basic statistical expression, it is not particularly helpful in itself for design work. For example, each annual maximum is deemed to be generated from a different unknown distribution, making distribution selection impossible. However, it is possible to obtain prediction expressions for HpT~magnitudes by making some reasonable assumptions concerning both the form of the F(x)t's and the nature of the changes in these distributions from one year to the next. A simple model for year-to-year change in the F(x)t's is suggested by the usual approach for trend-correcting recorded annual maxima. That is, the observed maxima are adjusted by adding or subtracting various amounts depending on some smoothing function passed through the data (where this "data" might be mean sea level values or the maxima themselves). The adjusted values are then analysed as if they were random variables generated from a common distribution. This procedure makes the implicit assumption that the unadjusted annual maxima arise from distributions which differ only in their degree of translation along the x-axis. In other words, any time trend in the annual maxima reflects the same trend in the location parameters of
the F(x)t's. The assumption of time-stability for all F(x)t parameters except location parameters is utilised in this paper. This seems a reasonable simplification
298
W.E. BARDSLEY ET AL.
provided that differences between individual F(x)t's arise mainly from differences in mean sea level. An implication of this basic assumption is that any trend in future mean sea levels will result in an exactly similar trend in the expected values of the annual maxima. Of course, a future upward trend in mean sea levels implies some form of climate change, which in turn suggests possible changes in the nature and frequency of storm surges. How such changes might alter the shape and scale parameters remains unknown. The approach utilised here therefore must be interpreted as a first-order approximation only. The method for generating prediction expressions can now be summarised. First, for trend-free data, proceed in the usual way by fitting a selected distribution to the recorded annual maxima. This will yield estimates of all unknown parameters. Alternatively, fit the distribution concerned to adjusted annual maxima if trend correction is required (making the assumption that the distributions of annual maxima differ only in location parameter values). When working with maxima which have been trend-adjusted (by whatever means), it is desirable to standardise the adjusted values to some selected year of record. This yields shape and scale parameter estimates, plus a location parameter estimate for the base year concerned. At this point, some checks should be made concerning the goodness of fit of the distribution concerned - using either a graphical method or calculating some formal test statistic. This applies to both trend-free or trend-corrected data. Given a reasonable fit, the next step is to obtain an estimate of the location parameter for year z (taken here to represent "the present" year). For a trendfree record of maxima, the required estimate is just the location parameter estimate for the distribution concerned. For trend-corrected maxima, the initial location parameter estimate by definition is with respect to the year selected for standardisation. This estimate can be converted to that of year z by using the trend previously established (making the assumption that any trend in maxima is a reflection only of a corresponding trend in location parameters). With the location parameter established for year r, any specified linear or non-linear trend in future mean sea levels will automatically define the sequence of location parameters of the F(x),'s to come. That is, if some future year t is specified to have a mean sea level greater by a certain amount d than year z, then the location parameter for year t is just that of z with d added. The mechanism of specification of future mean sea levels is not the concern of this paper. The mean levels may be derived from a deterministic climatic model, or possibly they may simply represent a specified "worst case" scenario for design purposes. At this stage, all the parameters are now either specified or estimated from data. These parameter values are now inserted into Eq. 2 to yield the required H,,T, estimates. This approach is quite general and need not even be confined
FUTURE SEA LEVEL EXTREMES UNDER CONDITIONS OF SEA LEVEL RISE
299
to parametric specification of the F(x)t's. For example, the parameter-free method of Bardsley ( 1988 ) could also be used here. EXTREME VALUE DISTRIBUTIONS
It is well known that any number of distributions can describe a given data set equally well, so we do not advocate any particular distribution or estimation procedure with respect to the F (x)t's. However, because the extreme value distributions have a long history in coastal design work, we give here their respective Newton's method calculation equations for obtaining the required x values in Eq. 2. The three extreme value distributions can be defined:
F(x)t=exp{-exp[- (x-~It)/O]}
Type 1
(3)
-b
where 0, ~ and a are (positive valued) scale parameters and c and b are (positive valued) shape parameters. The quantities q, to and ~ are location parameters, shown here with a t subscript to indicate their time-varying nature in the present context. The Newton iteration expressions for the Type 2 and Type 3 cases are given, respectively, in Eqs. 6 and 7 below as:
xj+l=xj~
~-blnp+ ~,f=r(Xj--tot)-b bE 7=~(xj_ to,)_(b+ 1)
(6)
a q n p + Y~L~(~t --xj) C Xj+l =xj 4
cy~L~(~t_xj)C_,
(7)
For the Type 1 (Gumbel) distribution, an explicit solution to Eq. 2 can be obtained:
HpT~=O-ln(-lnp)+ln[,~exp,
/a~
(8)
EXAMPLE
As an illustration of the kinds of results to emerge from the prediction method, an extreme value analysis was applied to a sequence of 60 annual sea level maxima obtained over the record period 1928-1987 at Port Adelaide
300
W.E. BARDSLEY ET AL.
(Outer Harbor), South Australia. A regression analysis of monthly means of recorded sea levels from this site indicated a linear increase in mean sea level of 2.85 m m per year. The original data were "raw" in that no attempt was made to isolate the meteorological component of the sea level values. The water level values were obtained from a stilling well, which d a m p e d out the influence of waves less than a few minutes in period. The 2.85 m m per year trend was assumed to apply to the maxima which were then adjusted by standardising with respect to the year 1928. We adopted this approach rather than directly testing the maxima themselves for a trend since such small changes would be very difficult to detect amongst the inevitable 'noise' of the extreme sea levels. However, such inferences from means to extremes do need to be treated with some caution (Walden and Prescott, 1983). As seen in Fig. 1, the trend-adjusted maxima on a Gumbel plot indicate an underlying curve with a slope decreasing with increasing y, where y = - l n [ - l n F ( x ) ] , and a numerical value for F(x) was estimated for each data point using the Weibull plotting position expression. This form of plot suggests the possibility of a Type 3 extreme value distribution (henceforth abbreviated to EV3 ). The suitability of the EV3 distribution for describing the data is supported by the reasonable fit of an EV3 prediction function:
x=~-aexp(-c-ly)
(9)
The particular prediction function shown as the curved line in Fig. 1 was obtained by three-point graphical fitting (Bardsley, 1989 ), yielding the parameter estimates of ~, a and c as 2.42 m, 0.71 m and 3.79, respectively. Although a reasonable data fit is achieved, we make the point again that we are not advocating any particular estimation procedure or distribution for 2"3
,
,
,
I
I
-0-75
0
0'75
1"50
2"25
,
,
c~2.1 I
t~ !
-1.50
3-00
3"75
Z+.50
H Fig. I. E x t r e m e v a l u e p l o t o f t r e n d - a d j u s t e d a n n u a l sea level m a x i m a , w i t h f i t t e d T y p e 3 e×t r e m e v a l u e p r e d i c t i o n f u n c t i o n . Site is O u t e r H a r b o r , P o r t A d e l a i d e ( A H D = A u s t r a l i a n H e i g h t D a t u m ).
301
FUTURE SEA LEVELEXTREMESUNDERCONDITIONS OF SEA LEVELRISE
general application. No doubt other distributions and estimation methods could be found which produce marginal improvements of fit to this data set. The location parameter value of 2.42 m is with respect to the standardised year 1928. On the basis of an annual increase in ~t of 2.85 m m per year since 1928, ~199o is obtained as 2.60 m. With z= 1990, subsequent ~t values were calculated on the basis of the intermediate future sea level rise given by Scenario II (NRC, 1987, p. 29). The values ofr~, ~ and ~, were then inserted into Eq. 7 to obtain H0.5T1990 through to T = 2100. The resulting curve is shown in Fig. 2, together with the upper bound line and the Scenario II curve for mean sea level at this location. More conservative design quantile curves such as H0.95 T1990 could also be added as required. The Ho.5T199o c u r v e in Fig. 2 shows an initial rapid increase reflecting the increase in the EV3 quantiles with increasing sample size. However, the rate of convergence to the limiting upper bound value is very slow and the curves come to be dominated by the specified sea level rise as time goes on. For design purposes, the upper bounds are best interpreted from a pragmatic viewpoint as a magnitude with an unspecified low probability of exceedance. Any upper bound value can only be an estimate of the true magnitude. Also, it would no doubt be possible to conceive of some combination of maximum surge and astronomical tide which would exceed the estimated bound for a given year. The upper bound is of course a consequence of the selection of an EV3 distribution, where the random variable cannot exceed ~t. For the EV3 or any other distribution possessing an upper bound, it is im-
ill . . . . . . . 3.0
<
up er bo~n6
2-0-
• e~T~ S
~
~
9redid e~ 1'5
1990
I
I
2020
I
I
I
I
I
2050 20 0 YEAR Fig. 2. Estimated upper bound and Ho.sT~99ofor sea level maxima at Port Adelaide, on the basis o f translated EV3 distributions. The mean sea level for the intermediate scenario is shown in the lower curve, raised by 1.7 m for graphing purposes.
302
W.E.BARDSLEYET AL.
portant to make the distinction between the HpT~and upper bound magnitudes with respect to time applications. The upper bound is with respect to the year concerned - that is, the highest water level in year T should not exceed the bound. On the other hand, the HpT~values state that there is a probability p of non-exceedance by the single highest sea level event between z and year T, without regard as to when that event occurs. CONCLUSION
The standard statistical methodology discussed here should prove useful as a simple means of obtaining practical design estimates under conditions of a rising sea level. If required, extension to the "true" maxima is achieved by simply redefining the annual maxima to be with respect to instantaneous water level. The design procedure should not be interpreted as a scientific forecast since too little is known at present about the factors influencing the future behaviour of mean sea levels. Rather, this technique is a means by which some selected scenario of mean sea level rise can be translated through to a scale of extreme event risk. The "factor of safety" in this case comes in at the start of the analysis with the selection of a rate of sea level rise which errs on the side of caution. ACKNOWLEDGEMENTS
For making available the sea level maxima used here, we are grateful to the Department of Marine and Harbours and the Coastal Branch of the Department of Environment and Planning (South Australia). REFERENCES Bardsley, W.E., 1988. Toward a general procedure for the analysis of extreme random events in the earth sciences. Math. Geol., 20:513-528. Bardsley, W.E., 1989. Graphical estimation of extreme value prediction functions. J. Hydrol., 110: 315-321. Blackman, D.L. and Graff, J., 1978. The analysis of annual extreme sea levels at certain ports in southern England. Proc. Inst. Civ. Eng., 65, Part 2: 339-357. Graff, J., 1979. Concerning the recurrence of abnormal sea levels. Coastal Eng., 2 (2): 177-187. Graff, J., 1981. An investigation of the frequency distributions of annual sea level maxima at ports around Great Britain. Estuarine Coastal Shelf Sci., 12: 389-449. Lennon, G.W., 1963. A frequency investigation of abnormally high tidal levels at certain west coast ports. Proc. Inst. Civ. Eng., 25: 443-449. Middleton, J.F. and Thompson, K.R., 1986. Return periods of extreme sea levels from short records. J. Geophys. Res., 91: (C10): 11,707-11,716. NERC (U.K.), 1975. Flood Studies Report, Vol. 1. Whitefriars Press, London. NRC (US), 1987. Responding to Changes in Sea Level (Engineering Implications). Natl. Academy Press, Washington, D.C.
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Suthons, C.T., 1963. Frequency of occurrence of abnormally high sea levels on the east and south coast of England. Proc. Inst. civ. Eng., 25: 443-449. Walden, A.T. and Prescott, P., 1983. Identification of trends in annual maximum sea levels using robust locally weighted regression. Estuarine Coastal Shelf Sci., 16:17-26. Walden, A.T., Prescott, P. and Weber, N.B., 1981. Some important considerations in the analysis of annual maximum sea levels. Coastal Eng., 4(4): 335-342.