Extreme Sea Levels Philip L. Woodworth Permanent Service for Mean Sea Level with thanks to David Pugh, David Blackman and Roger Flather.

Extreme Sea LevelsExtreme Sea Levels

Philip L. WoodworthPhilip L. WoodworthPermanent Service for Mean Sea LevelPermanent Service for Mean Sea Level

with thanks to David Pugh, David Blackman and Roger with thanks to David Pugh, David Blackman and Roger FlatherFlather

ContentsContents

• IntroductionIntroduction

• Annual Maxima methodAnnual Maxima method

• Joint Probability methodJoint Probability method

• Complementary value of tide gauge data and Complementary value of tide gauge data and numerical modellingnumerical modelling

• Changes in extremes with climate changeChanges in extremes with climate change

INTRODUCTION

Coastal planners need to know the risk of flooding to structures such as houses, factories and power stations at the coast so that decisions can made on where to site them and protect them.

High water extreme events typically result from a high water on a spring tide and a storm surge.

Let Q(z) be the Probability of a level z being exceeded in any 1 year.

(Don’t worry for now about how to calculate Q(z)).

Then the RETURN PERIOD T(z) = 1/Q(z) is the average time between which levels higher than z occur.

The DESIGN RISK is the Probability that a given z will be exceeded during the design life (D years) of the structure.

If Q(z) is the Probability of exceeding z in 1 year then

(1 – Q(z)) is the Probability of NOT exceeding z in 1 year

(1 – Q(z)) 2 is the Probability of NOT exceeding z in 2 years

(1 – Q(z)) D is the Probability of NOT exceeding z in D years

Then DESIGN RISK = 1 - (1 – Q(z)) D

e.g. from next figure we see that if engineers build a structure to a DESIGN RETURN PERIOD T(z) of 100 years, then if the structure is required to exist for D = 100 years, there will be a 63.5 % chance of the level z being exceeded at some occasion in that time.

Relationship between the risk of encountering an extreme sea level with aReturn Period of 100 years in the lifetime of the structure, as a function ofthe intended period of operation of the structure

Note that houses, power stations etc. at the coast all have D = 100 years or thereabouts. To get a small DESIGN RISK of being flooded in that time, we have to make the design return period T(z) as large as possible.

For nuclear power stations, the design T(z) may be 100,000 or 1,000,000 years. In the Netherlands, houses are constructed with T(z) of 10,000 years. In the UK, T(z) is often as low as 1,000 years.

e.g. if D = 100 years, and design risk is required to be just 0.1 (10% chance of flooding at some time during the 100 years) then a design return period T(z) of 950 years is needed.

(Use the equations on previous pages to check this.)

To calculate Q(z) for a range of values of z, we can use:

• Tide gauge data plus statistical models

• Numerical modelling information (plus statistical models)

• Tide gauge data and numerical modelling in combination

In the following examples we shall use tide gauge data from Newlyn.

Example data used here are taken from Newlyn(Mean Tidal Range 3 m)

ANNUAL MAXIMA METHOD

We have 84 complete years of Newlyn data (within 1916-2000), so we have 84 ANNUAL MAXIMUM water levels. These are histogrammed on next figure.

Note that Highest Astronomical Tide (HAT) (which is at z=3.0 m) was exceeded only 28 times, because in most years the astronomical tide did not approach HAT and the surges at high water were not big enough to take the combined level over HAT.

Curve shows Q(z) = Probability of z being exceeded in any 1 year

e.g. Q(z=HAT) = 0.33 = 28/84

The Q(z) can be plotted against z for values of z for which we have data. This is called an ‘Extreme Level Distribution’.

Alternatively, and more normal, is to plot the distribution in another way: z versus log(T(z)). The use of log(T) is such that it makes the resulting curve approximately linear – see next figure.

This curve can be parameterised easily for interpolation, but that does not help if we need to extrapolate it in order to estimate the z values corresponding to higher T(z) values (or, if you prefer, very low probabilities Q(z)).

To perform the extrapolation we need to assume one of the Generalised Extreme Level (GEV) family of curves.

The GEV family of curves is derived from the shapes of the extremes of Gaussian- (or Normal-) type distributions and have the form:

z = b + a (1 – e –kX)

where z is the level of interest and X = log(T(z))

In the previous example, k > 0.0. The special case k=0 is called a Gumbel Distribution and sometimes the GD is preferred as a simpler choice of curve to fit than the GEV curve which has the extra parameter (k).

Once the GEV (or GD) curves have been fitted by least-squares (or, more usually these days, maximum likelihood) to the available data, then the curves can be extrapolated to larger T(z) values.

In practice, one can extrapolate out to values of T(z) which are approximately several times the record length (i.e. several times the 84 years in this example for Newlyn).

Software now exists which can perform such calculations easily and produce formal errors on estimates of z corresponding to extrapolated T(z) values. It is very important to know such errors.

Joint Probability Method

The Annual Maxima method is ‘wasteful’ in that it uses tide gauge information from only the highest high waters each year. This ignores all the other data from the rest of the year, which is a bit crazy!

The JPM uses the fact that the statistics of the tide and of surges are largely independent (not completely true) and compiles separate tables of the distributions of both quantities. So, we can learn about the statistics of large positive surges even if they occur at low water, for example; in the Annual Maxima method such surges would not have contributed to the analysis.

An advantage of the JPM is that it allows to estimate much smaller probabilities from the data alone, without need for the gross extrapolations of the Annual Maxima method. Also much shorter data sets can be used than in the AM method e.g. even 4 years might be useful compared to the 84 from before.

The first step is to perform a tidal analysis (e.g. from the TASK-2000 package) such that the time series of (usually hourly) sea level values for the year is divided into ‘tide’ and ‘surge’ time series.

The tidal series has a height frequency distribution as shown on the next page. The surge time series will have a distribution which is approximately Gaussian.

The first step is to perform a tidal analysis such that the time series of (usually hourly) sea level values for the year is divided into ‘tide’ and ‘surge’.

The tidal series has a height frequency distribution as shown on the next page. The surge time series has a distribution which is approximately Gaussian.

Then, inside a computer of course, we can make a 2-dimensional table which is like a 2-D version of the histogram used above for the Annual Maxima method.

The following page shows a highly schematic example of the table, in practice many more rows and columns would be used. But mostly we need only consider the higher tide rows which have a chance of contributing to an overall high water extreme.

Normalised frequency distributions for tide (vertical axis) and surge (horizontal axis). Surge = 0.1 for example means surge between 0.05 and 0.15 m.

-0.2 -0.1 0.0 0.1 0.2

3.2 0.1 .01 .02 .04 .02 .01

3.1 0.2 .02 .04 .08 .04 .02

3.0 0.3 .03 .06 .12 .06 .03

2.9 0.3 .03 .06 .12 .06 .03

2.8 0.1 .01 .03 .04 .02 .01

A total level of 3.4 m (i.e. between 3.35 and 3.45) would be obtained 11% of the time (of the high tidal levels represented in the table) from tide+surge 3.2+0.0 (0.04), 3.1+0.1 (0.04) and 3.0+0.2 (0.03)

The statistics included in this table can be converted into Q(z) and T(z) form similar as for the Annual Maxima method enabling similar Extreme Level Distribution plots to be produced.

For more details, see Pugh (1987) book

WHAT CAN YOU DO IF YOU HAVE NO TIDE GAUGE DATA FROM A LOCATION WHERE YOU WANT TO HAVE

EXTREME LEVEL INFORMATION?

• Simple regional approach methods

• Sophisticated ‘spatial approach’ modelling of Coles and Tawn

• Use numerical tide+surge models

Example of simple methods, where you have data, define:

α100 = 100-year return water level

-------------------------------------------

(HAT + 100-year return surge level)

If the large surges always occurred at high astronomical tide, then this quantity would be 1.0. In practice of course, they do not always, so it is often much less than 1. Around the UK it is typically 0.8, falling to 0.7 in the southern North Sea where tide-surge interaction luckily causes surges to avoid high water.

Once values of α100 have been acquired for an area, then it may be possible to use the same value at sites where there are no good surge data (but some basic knowledge of the tide is still needed).

WHAT CAN YOU DO IF YOU HAVE NO TIDE GAUGE DATA FROM A LOCATION WHERE YOU WANT TO HAVE

EXTREME LEVEL INFORMATION?

• Simple regional approach methods

• Sophisticated ‘spatial approach’ modelling of Coles and Tawn

• Use numerical tide+surge models

POL NISE model grid (~12km) - nested in NEAC

Storm surge extremes – numerical model approachModel runs forced by long met data sets produce realistic surge climatology - which can then be analysed like tide gauge observations.

Two model runs are usually carried out for:

1. tide + met (air pressure and wind) forcing2. tide only

model fields stored hourly then 1 – 2 gives the storm surge component.

Data can then be used for Annual Maxima or JPM as for tide gauge data, or be used with the gauge data as an interpolation tool.

OTHER EXTREME LEVEL TECHNIQUES

• ‘r largest’ method rather than the ‘1 largest’ method of Annual Maxima

• Revised JP Method

• Peaks over threshold

• Percentiles

Some warnings about all methods:Some warnings about all methods:

• The methods are designed for mid-latitude climates where The methods are designed for mid-latitude climates where extremes come from winter storms. extremes come from winter storms.

• Experience is needed in dealing with data sets which have large Experience is needed in dealing with data sets which have large outlier extremes. It is important to decide if they are outlier extremes. It is important to decide if they are representative or not, as they affect analysis results representative or not, as they affect analysis results considerably.considerably.

• None of the methods work well for ‘really extreme’ events e.g. None of the methods work well for ‘really extreme’ events e.g. tsunamitsunami

• We have discussed extreme still water levels (tides + surges) We have discussed extreme still water levels (tides + surges) only. Extreme waves, and tide-surge-wave interactions, also have only. Extreme waves, and tide-surge-wave interactions, also have to be considered. And extreme waves + currents for off-shore to be considered. And extreme waves + currents for off-shore industry.industry.

• Refs. Pugh Refs. Pugh Tides, surges and mean sea-levelTides, surges and mean sea-level, 1987, chapter 8; , 1987, chapter 8; UK MAFF reports obtainable from UK MAFF reports obtainable from http://www.pol.ac.uk/ntslf/http://www.pol.ac.uk/ntslf/

TsunamiTsunami

Scenario:

Cumbre Vieja volcano, La Palma, Canary Islandsslides into the sea

Tsunami waves O(5-10m)hit NW European Shelf.

(Picture from Benfield Greig Hazard Research Centre, UCL)

Some warnings about all methods:Some warnings about all methods:

• The methods are designed for mid-latitude climates where The methods are designed for mid-latitude climates where extremes come from winter storms. extremes come from winter storms.

• Experience is needed in dealing with data sets which have large Experience is needed in dealing with data sets which have large outlier extremes. It is important to decide if they are outlier extremes. It is important to decide if they are representative or not, as they affect analysis results representative or not, as they affect analysis results considerably.considerably.

• None of the methods work well for ‘really extreme’ events e.g. None of the methods work well for ‘really extreme’ events e.g. tsunamitsunami

• We have discussed extreme still water levels (tides + surges) We have discussed extreme still water levels (tides + surges) only. Extreme waves, and tide-surge-wave interactions, also have only. Extreme waves, and tide-surge-wave interactions, also have to be considered. And extreme waves + currents for off-shore to be considered. And extreme waves + currents for off-shore industry.industry.

• Refs. Pugh Refs. Pugh Tides, surges and mean sea-levelTides, surges and mean sea-level, 1987, chapter 8; , 1987, chapter 8; UK MAFF reports obtainable from UK MAFF reports obtainable from http://www.pol.ac.uk/ntslf/http://www.pol.ac.uk/ntslf/

Changes of Extremes and Risk with Changes of Extremes and Risk with Climate ChangeClimate Change

• Simple approach which considers just a MSL Simple approach which considers just a MSL change and resulting changes in z vs. T(z)change and resulting changes in z vs. T(z)

an order of magnitude increase in risk at an order of magnitude increase in risk at NewlynNewlyn

• Complex approach which models changes of MSL, Complex approach which models changes of MSL, tides, surges etc. in a future climatetides, surges etc. in a future climate

conclusions are very dependent on confidence conclusions are very dependent on confidence in global climate modelsin global climate models

Changes of Extremes and Risk with Changes of Extremes and Risk with Climate Change:Climate Change:

• Simple approach which considers just a MSL Simple approach which considers just a MSL change and resulting changes in z vs. T(z)change and resulting changes in z vs. T(z)

an order of magnitude increase in risk at an order of magnitude increase in risk at NewlynNewlyn

• Complex approach which models changes of MSL, Complex approach which models changes of MSL, tides, surges etc. in a future climatetides, surges etc. in a future climate

conclusions are very dependent on confidence conclusions are very dependent on confidence in global climate modelsin global climate models

Integrated effects of Integrated effects of climate change on UK climate change on UK

coastal extreme sea levelscoastal extreme sea levels

(As an example of such a (As an example of such a ‘complex approach’ and with ‘complex approach’ and with a suspicion that ‘things are a suspicion that ‘things are

getting worse’)getting worse’)

Floods in the IoM 2002Floods in the IoM 2002Douglas

Ramsey

"the worst in living memory"

£4m damage in 3 hours

Pictures from http://www.iomonline.co.im/ftpinc/weather/febhightide.asp.

AimAim• To derive insight into on changes and To derive insight into on changes and

trends in extreme sea levels from trends in extreme sea levels from existingexisting informationinformation

• Changes in extreme SL at the coast result Changes in extreme SL at the coast result

from:from:a) global MSL change + regional variationsa) global MSL change + regional variationsb) regional land movementsb) regional land movementsc) tidal changes due to increased SLc) tidal changes due to increased SLd) changes in storm surges due to d) changes in storm surges due to

changes in "storminess"changes in "storminess"

a) MSL changea) MSL change

• UK mean sea level (MSL) UK mean sea level (MSL) is risingis rising

• Plot shows MSL "relative" Plot shows MSL "relative" (to the land) as (to the land) as measured by tide measured by tide gaugesgauges

• Corrected for local land Corrected for local land movements, the movements, the "absolute" MSL trend is "absolute" MSL trend is about +1mm/y = about +1mm/y = 10cm/century10cm/century

• IPCC predicts +47cm by IPCC predicts +47cm by 21002100

b) Land movementsb) Land movements• Land subsidence or uplift Land subsidence or uplift

can result from: can result from: – post-glacial reboundpost-glacial rebound– water extractionwater extraction– sediment compaction sediment compaction

etc.etc.

• Estimates (mm/y) based Estimates (mm/y) based on geological data on geological data (Shennan, 1989) are (Shennan, 1989) are shown hereshown here

• Recent results (Shennan, Recent results (Shennan, in press) are not includedin press) are not included

c) Tidal changesc) Tidal changes• Tides are modified Tides are modified

by SL riseby SL rise

• Increased depth Increased depth longer wavelengthlonger wavelength

• Figure shows the Figure shows the change in MHW change in MHW due to an assumed due to an assumed 50cm rise in MSL50cm rise in MSL

• Changes at the Changes at the coast are 45 - coast are 45 - 55cm55cm

d) Extreme storm d) Extreme storm surgesurge• Computed change in 50-Computed change in 50-

year surge elevation year surge elevation

"2"2COCO22"-"control""-"control"

• Produced from 30-y Produced from 30-y runs of surge models runs of surge models forced by met data from forced by met data from ECHAM4 T106 time-slice ECHAM4 T106 time-slice expts.expts.

• Caution! Similar studies Caution! Similar studies with other climate with other climate GCMs, different GCMs, different sampling and extreme sampling and extreme value analysis give value analysis give different results.different results.

(from STOWASUS-2100 EU ENV4-CT97-0498)

Change in Change in relativerelative extreme SLextreme SL• Taking the sum of Taking the sum of

changes in changes in MSL + MHW MSL + MHW + S50 + land + S50 + land movement movement (Scottish uplift will (Scottish uplift will decrease the change)decrease the change)

• we obtain change in we obtain change in extreme sea level (cm) extreme sea level (cm) relative to the landrelative to the land for for 2075 shown in the plot2075 shown in the plot

• Caution! - uncertainty Caution! - uncertainty in each componentin each component

Rate of change of Rate of change of relativerelative SL SL

• Assuming the Assuming the changes in changes in relativerelative extreme SL occur extreme SL occur between 1990 and between 1990 and 20752075

• Mean rates are Mean rates are shown … c.f. shown … c.f. official UK advice official UK advice (boxed numbers)(boxed numbers)

Coastal areas at riskCoastal areas at risk

• Areas below Areas below 1000-year return 1000-year return period levelperiod level

• By 2100:By 2100:thethe1 in 1000-y1 in 1000-y level level may become a may become a 1 in 1 in 100-y100-y level level

ConclusionsConclusions

• Some of the methods used to compute extreme Some of the methods used to compute extreme levels have been described but see refs. for more levels have been described but see refs. for more details.details.

• Also a case study of possible changes in extremes Also a case study of possible changes in extremes around the UK has been described around the UK has been described we suggest we suggest that other countries conduct similar studies.that other countries conduct similar studies.

• Note that the IPCC Third Assessment Report Note that the IPCC Third Assessment Report discussed extensively changes in MSL, but pointed discussed extensively changes in MSL, but pointed out that it is primarily the extreme events which do out that it is primarily the extreme events which do damage, and that far more study is required than damage, and that far more study is required than has been made so far on extremes and on their has been made so far on extremes and on their possible changes in future. possible changes in future.

• So the GLOSS community must include this topic in So the GLOSS community must include this topic in its programme of work.its programme of work.