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Estimating extreme wave height probabilities from observations and the ERA 40 reanalysis

A weather forecast is essentially an initial value problem. Given the atmospheric state (the ‘weather’) at one time, the state at a later time can be calculated.

The problem, however, with this simple view is that the atmospheric state is never known completely. For large parts of the atmosphere observations are not available (remote areas, upper air), and available measurements inevitably contain errors. To overcome this problem, the initial state of a new weather forecast is obtained by a combination of the latest forecast and all new observations. The latest forecast has usually been initialised six hours earlier and gives a good first guess for the initialisation of a new forecast.

Most importantly, it provides a complete description of the atmosphere as by definition it has values of all relevant quantities at all grid points. The first guess is then combined with the newly available observations in a way respecting physical laws. The observations ‘push’ the first guess towards ‘reality’. This step, by no means trivial, is called analysis. At ECMWF the analysis costs about half of the total CPU-time needed to make a 10 day forecast, the other half being used for the time integration.

As a consequence, operational forecast centres naturally produce a complete description of the atmosphere's state, usually four times a day. However, weather forecast models and analysis procedures are continually improved. Variability in the operational analyses is dominated by these changes rather than by natural variability, making them unsuitable to study long-term changes. The aim of re-analysis is to overcome this problem of inhomogeneity. A state-of-the-art analysis system is used to repeat the analysis procedure for the past. As a result one obtains a complete description of the atmosphere over a long period of time, which is free of inhomogeneities due to model changes. Unfortunately, inhomogeneities due to changes in data coverage remain1).

ERA 40

The first global reanalyses were produced in the first half of the 1990s2,3). They are widely used in climate and meteorological research. Progress in modelling and data assimilation as well as the availability of new data sets led ECMWF to conduct a new reanalysis, ERA 40, covering the 45 years from September 1957 to August 2002. It uses a version of ECMWF’s Integrated Forecasting System (IFS) that was operational in June 2001, albeit on a coarser grid (+/-125 km instead of +/-40 km).
A large subset of the complete ERA-40 data set is available at http://data-portal.ecmwf.int/data/d/era40_daily/ on a 2.5° x 2.5° grid.

A distinguishing feature of the IFS is that over the oceans the surface roughness depends on the sea state4,5), and the sea state is obtained from the WAM wave model6). Thus wave information is a natural product of ERA 40. One of the most important wave parameters is the significant wave height (HS), a measure of the severity of the sea state. (To be precise it is the 20 minute-average of the upper third of a wave height record.)The length of the ERA 40 data set makes it especially suitable to study variability and extremes of weather-related quantities. Information about decadal variability of climate quantities and their extremes is of great interest for climate (impact) research. An example of an extreme parameter is the 100-year return wave height (H100). This is the significant wave height that on average is exceeded only once every 100 years. It is used in the design of ships and of maritime structures.

Validation and correction of the ERA-40 wave product

We have thoroughly validated the raw ERA 40 wave data against buoy and altimeter data. Buoys provide high-quality continuous point measurements at a very limited number of sites. Satellite-born altimeters provide near-global coverage, but every point is sampled only once in several (typically 10) days.

Figure 1 shows the time series of HS as derived from measurements at buoy 46001 in the Gulf of Alaska (148.3°W, 56.3°N) during 1988 (red), together with the corresponding ERA 40 data (blue). Three properties of the ERA-40 data can easily be recognized: (a) the two curves are nearly perfectly in phase, (b) low wave heights tend to be slightly overestimated by ERA 40, and (c) high waves tend to be substantially underestimated. These three features are not a peculiarity of the special location, but a general property of the ERA 40 wave data. Among the reasons for these deficiencies are spatial resolution (P. Janssen, pers. communication) and a slight underestimation of high wind speeds7).

Figure 1: Measured (green) and modelled (blue) HS at buoy 46001 (148.3°W, 56.3°N) in 1988.
Figure 1: Measured (green) and modelled (blue) HS at buoy 46001 (148.3°W, 56.3°N) in 1988.

Apart from the underestimation of large wave heights ERA 40 waves also suffer from inhomogeneities due to changes in the data that were assimilated. As a synthesized picture of the data Figure 2 shows the time series of the globally averaged monthly mean Hs from ERA 40 (blue). From 1991 onwards wave height data from altimeters flown onboard satellites became available and were assimilated. The impact of these data is clearly seen, especially for the period from December 1991 to May 1993, when erroneous data were used. Another inhomogeneity is visible in 1996, when the altimeter data changed from ERS 1 to ERS 2.

Fortunately, it was possible to devise a non-parametric correction method for the ERA 40 data.
A corrected dataset was created8) which has no bias with respect to altimeter-based wave height retrievals and which is free of obvious inhomogeneities resulting from differences in wave-height data that were assimilated (Figure 2, red). Furthermore, reliable estimates of the 100-year return wave height could be obtained from the raw ERA 40 data by a calibration against buoy measurements. These and other results from the ERA 40 wave data have been incorporated into the web-based KNMI/ERA 40 Wave Atlas.

Figure 2: Time series of the monthly mean, globally averaged HS from the raw (blue) and the corrected (red) ERA 40 data. Monthly means are computed from the 6 hourly fields.
Figure 2: Time series of the monthly mean, globally averaged HS from the raw (blue) and the corrected (red) ERA 40 data. Monthly means are computed from the 6 hourly fields.

Estimation of extreme significant wave heights

For safety considerations it is important to know extreme wave heights, i.e., wave heights that are, on average, exceeded only once per 20, 50, or 100 years. The ERA 40 data set has proved an invaluable basis to derive global estimates of these extremes. Note that extremes of significant wave height rather than those of individual waves are obtained. Estimates of the 100-year return significant wave height H100 are obtained using the Peak-Over-Threshold (POT) method9), in which the tail of the wave height distribution is fitted to the Generalized Pareto Distribution (GPD), the limit distribution for extremes.

Estimating H100 both from buoy measurements and from the raw ERA 40 data yields a linear relationship between the two10),
H100(buoy) = 0,52 + 1,30 H100(ERA 40). (1)

This relation is illustrated in Figure 3.

Figure 3: Linear correlation between 100 year return value estimates of HS from buoy data and from ERA 40. The dashed line is eq. (1).
Figure 3: Linear correlation between 100 year return value estimates of HS from buoy data and from ERA 40. The dashed line is eq. (1).

Buoy locations are very sparse and unevenly distributed in space, and the largest value of H100 found at the buoy locations is about 17 m (Figure 3). Therefore it would be preferable to have a relation between H100 estimates from ERA 40 and from satellites, respectively. However, satellites cross a given point only once in typically 10 days. Together with the relative shortness of the satellite record this gives too few data for a reliable extreme-value estimate at a given location. However, as far as parts of the estimation procedure were possible with satellite data their results are not incompatible with (1). We therefore apply this equation globally and for all values of H100.

Figure 4 shows the H100 values obtained by applying the POT method to the ERA 40 data and correcting the results using (1). It is obvious that the highest values occur in the North Atlantic. While mean wave heights are not higher in the North Atlantic than they are in the North Pacific or in the Southern Ocean (Figure 5), the North Atlantic shows the highest variability (not shown). In other words, conditions in the Southern Ocean are always rough, while in the North Atlantic you can be lucky and the sea is calm even in winter, or you find yourself between the highest waves possible on earth.

Figure 4: 100 year return HS from ERA-40, corrected using the relationship displayed in Figure 3. Note that the results pertain to averages over 1.5° x 1.5° and 6 hours, that shallow water effects are not included, and that tropical cyclones are not resol
Figure 4: 100 year return HS from ERA-40, corrected using the relationship displayed in Figure 3. Note that the results pertain to averages over 1.5° x 1.5° and 6 hours, that shallow water effects are not included, and that tropical cyclones are not resol

Besides an estimate based on the whole ERA 40 period, Figure 4 also contains estimates of H100 for three different 10-year periods. The estimates obtained from these periods differ in the Northern Hemisphere storm tracks. Specifically, the estimates in the North Pacific storm track region have increased, and in the North Atlantic the pattern has changed. These differences can be attributed to decadal variability in the Northern Hemisphere, especially to changes in the phase of the NAO8). This example shows that it is important to take account of climate changes when designing maritime structures. A more detailed investigation reveals that in the North Atlantic changes in estimates of H100 are due to changes in the intensity of storms, while in the Southern Hemisphere they are mainly due to changes in the number of storms. In the North Pacific both factors contribute.

Summary and Conclusions

The ERA 40 reanalysis carried out at ECMWF produced 45 years (September 1957 - August 2002) of data describing the state of the atmosphere and the ocean surface four times a day.
A thorough assessment of the ERA 40 wave height data revealed that they (a) capture very well the variability of the true wave heights on all time scales, (b) slightly overestimate low wave heights, and (c) severely underestimate high wave heights. Furthermore, inhomogeneities due to the assimilation of different data sources are clearly present. A non-parametric correction method was devised that eliminates most of these problems.

Figure 5: January (left panel) and July (right panel) averages of HS (colours) and propagation direction of waves (arrows) for the whole ERA 40 period.
Figure 5: January (left panel) and July (right panel) averages of HS (colours) and propagation direction of waves (arrows) for the whole ERA 40 period.

Despite the underestimation of high wave heights it is possible to give reliable estimates of extreme significant wave heights (‘100-year-return values’). Estimates based on the raw ERA 40 wave data and those from buoy measurements reveal a linear relationship that can be exploited to obtain global reliable return value estimates based on the ERA 40 data. Maps of the ERA 40 data and derived quantities can be found in the web-based KNMI/ERA 40 Wave Atlas.

Acknowledgments

We are indebted to many persons for their help and pleasant collaborations. Jean-Raymond Bidlot and Peter Janssen provided valuable suggestions and comments and helped with advice. Sakari Uppala and Per Kållberg as leaders of the ERA 40 production team where always open to our comments and provided valuable help in dealing with the technical aspects of the ERA-40 system. Helen Snaith helped with the altimeter data. The buoy data were obtained from NDBC-NOAA. This work was funded by EU as part of the ERA-40 project (no. EVK2-CT-1999-00027).

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