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Global and Planetary Change 57 (2007) 139 – 150 www.elsevier.com/locate/gloplacha Mapping the return periods of extreme sea levels: Allowing for short sea level records, seasonality, and climate change N.B. Bernier a,⁎, K.R. Thompson a , J. Ou a , H. Ritchie b b a Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4J1 Meteorological Service of Canada, 45 Alderney Drive, Dartmouth, Nova Scotia, Canada B2Y 2N6 Available online 25 January 2007 Abstract This study of extreme sea levels is motivated by concern over increased coastal erosion and flooding under plausible climate change scenarios. Extremal analyses are performed on the annual and seasonal maxima from 24 tide gauge stations in the Northwest Atlantic. At data poor locations, a 40 yr surge hindcast and information from short observation records are used to reconstruct sea level records prior to the annual and seasonal analysis of extremes. A Digital Elevation Model is used to generate spatial maps of the return period of extreme sea levels associated with specified flooding probabilities under current conditions and under projected global sea level rise scenarios for the next century. It is the first time such maps have been produced. Their primary advantage is that extreme sea levels are expressed in terms of inundated areas as opposed to a critical flood value about an arbitrary datum. Another novel aspect of this study is that the extremal analyses are carried out for specific seasons. © 2006 Elsevier B.V. All rights reserved. Keywords: sea level; extremal analysis; storm surges; spatial mapping of coastal flooding; Digital Elevation Model; climate change 1. Introduction Over coming decades, projections of global sea level rise and plausible climate change scenarios (e.g., Houghton et al., 2001) point to an increased flooding risk for many low lying coastal regions. This is particularly true for areas where subsidence of the earth's crust combines with global sea level rise to enhance the rate of relative sea level rise (i.e., with respect to a fixed land point). In the southern Gulf of St. Lawrence the projected change of sea level over the next century is of order 1 m (Bernier and Thompson, 2006). This creates serious concern for low lying coastal areas ⁎ Corresponding author. E-mail address: [email protected] (N.B. Bernier). 0921-8181/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.gloplacha.2006.11.027 already threatened by instantaneous flood events of the same magnitude. Unfortunately, many of the regions at risk have insufficient sea level data available for standard extremal analysis. It is therefore difficult to evaluate the distribution of extremes under current conditions and even more problematic to assess the effect of climate change on the return period of extremes. In this study we map the return period of extreme sea levels using (i) hourly sea level observations, (ii) a dynamically based surge model, and (iii) a Digital Elevation Model (DEM). Following in the footsteps of Pugh and Vassie, 1980, we pay particular attention to the case of short sea level records (e.g., several years in length). The surge model has already been extensively validated for short term forecasts (e.g., Bernier and Thompson, 2006). The purpose here is to use the model to generate a 40 yr surge hindcast that can be used to 140 N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 calculate extreme statistics, even for locations with few sea level observations. Thus, our interest is not in short term surge forecasting but rather the calculation of extreme statistics, essential information for the development of any sensible adaptation strategy to global sea level rise and climate change. The DEM is used to downscale the flooding statistics to the community level (i.e., to downscale the results of the extremal analysis to a horizontal resolution of meters). Maps of the return period of extreme flood levels can then be used by local authorities to quickly identify lands that will be flooded on average every 2, 10, or even 100 yr. With this type of information, communities can, for example, readily identify areas where development should be prohibited or the transport network protected. The DEM maps of return period can also be used by ecologists to evaluate the impact of floods on the survival of species at risk. The piping plover is an example of a species at risk. It is found along the shorelines of the Northumberland Strait, the Magdalen Islands and some beaches of Nova Scotia, Prince Edward Island and Newfoundland. In 2001, there were about 220 pairs of plovers and only 43 single birds left (source: Environment Canada Species at Risk: http:// www.speciesatrisk.gc.ca/). Piping plovers arrive at their nesting grounds in Atlantic Canada in late April. In addition to predators and human disturbance, loss of habitat due to storm surges and sea level rise is considered a major threat to their survival. The important point with respect to the present study is that the return periods of extreme sea levels are required for only one part of the year: the breeding season. In this paper we show how to allow for seasonality in the extremal analysis. The structure of the paper is as follows. In Section 2, the observed sea level records, standard extremal analysis, and the reconstruction of sea level records in data poor locations are described. Extremal analyses are tailored and performed on seasonal maxima as opposed to the traditional annual maxima. In Section 3, the results of the seasonal extremal analyses are downscaled to the community/ecosystem level using a recently acquired and prepared DEM. In Section 4, projections of the return periods of extreme sea levels are made under climate change scenarios that include changes in the severity of storms. The results of the study are summarized and discussed in Section 5. 2. Extremal analysis of observed and hindcast sea levels In this section, extremal analyses are performed on the annual maxima of hourly sea levels in order to estimate the return period associated with critical flood Fig. 1. Map of the domain of the storm surge model. The numbers mark the locations of the tide gauges used to validate the storm surge model and the reconstructed total sea level. Station 14, marks the location for which a Digital Elevation Model is available. N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 levels (ηc). The sea levels used in this study are for 24 gauges located along the east coast of the United States, Quebec, and Atlantic Canada (Fig. 1). Two types of hourly sea level were used: observations and reconstructed records using surge model hindcasts. 2.1. The observed sea levels The observed hourly sea level records for the 24 locations were subject to thorough quality control prior to analysis, including visual inspection of all large recorded values. All records used in this study have a minimum of 10 yr of observations between 1960 and 1999. The long term means were removed from all records prior to analysis. Estimates of the frequency of coastal flooding are typically based on the analysis of annual maxima (e.g., Gumbel, 1958; Leadbetter et al., 1983; Coles, 2001). Extreme value theory assumes that the variable we are studying, Mn, is the maximum of n independent, identically-distributed (iid) random quantities: Mn ¼ maxfg1 ;: : : ; gn g ð1Þ It is further assumed that sequences of location and scale parameters (an and bn N 0) exist such that Pr 141 (Mn b an + bnηc) converges to a non-degenerate function as n tends to infinity. Under these conditions the probability distribution function of Mn is one of three extreme value types (e.g., Coles, 2001). The Type I distribution is commonly used in the study of sea level records (e.g., Dixon and Tawn, 1999; Lowe et al., 2001; Woodworth and Blackman, 2002; Bernier and Thompson, 2006). The Type I extremal distribution has the form PrðMn bgc Þ ¼ expf−exp½−ðgc −an Þ=bn g ð2Þ In this study, maximum likelihood is used to estimate the location and scale parameters from the annual and seasonal maxima of the hourly sea level observations. The application of the theory is illustrated for Charlottetown, one of the few locations in the study region with a long (1938–present) and a fairly complete observation record. The annual maxima and minima (about the annual mean) were first extracted from the record (top panel of Fig. 2). Note the difference in the interannual variability of the maxima and the minima and the offset between the mean sea level and the extremes. The latter reflects the effect of the tides at Charlottetown. The maxima and minima were ordered and plotted on Type I probability paper. The dots show the annual Fig. 2. Observed sea levels at Charlottetown. The top panel shows the annual mean about the long term mean in meters. Note the trend in annual means. The adjusted annual maxima and annual minima are also plotted. The average difference between the means and maxima and minima is due in large part to the tides. The bottom two panels are Type I plots of the maxima and minima about the annual means. The x-axes are the return period in years and the y-axes are the return level in meters. The dots are the ordered, observed adjusted annual maxima (bottom left panel) and observed annual minima (bottom right panel) from 1938 to 2004. 142 N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 maxima (bottom left panel of Fig. 2) and annual minima (bottom right panel of Fig. 2). The lines were fit using maximum likelihood. The shaded area marks the 95% confidence intervals obtained using the delta method (e.g., Coles, 2001). Note the difference in the slope of the return level line (e.g., the difference between the 2 yr and the 40 yr return period) of the maxima and minima. The largest differences in the slopes of maxima to minima are found at the southern most stations where hurricanes are known to lead to extreme positive residuals. Other factors such as an increase in frictional effects associated with a decrease in water depth during large negative residual events can limit their amplitudes (e.g., Grant and Madsen, 1979). 2.2. The hindcast sea levels An important limitation of standard extremal theory is that it requires the hourly sea level records exceed about 30 yr in length. Unfortunately this is rarely the case. In Atlantic Canada, only a small number of such records is available. For the Gulf of St. Lawrence, a region that regularly sustains damage due to large flood events, only one such record is available. Another problem is that the distribution of both tides and surges varies considerably over the region. Knowledge of the distribution of extremes at one location is therefore not a good estimate of the distribution of extremes at other locations. Fig. 3. Evolution of the surge event of December 1972. The top 6 panels are snapshots of the hindcast surges taken 9 h apart. The colorbar is surge level in meters. The bottom panel shows the observed filtered residuals (black) and the hindcast surges (red) at the gauge location marked by the yellow dot. The series cover two weeks surrounding the surge event of 17 December 1972. The vertical lines mark the times at which the snapshots are taken. N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 Our solution to this problem is to use a validated storm surge model to reconstruct long sea level records in data poor regions. The return period of seasonal extreme sea level are then calculated from the seasonal maxima of the reconstructed records. The details of the surge model, and its validation, are given in Appendix A. The hindcast skill of the model is illustrated in Fig. 3. The top 6 panels are a sequence of snapshots of sea surface height. The bottom panel is a comparison of the hindcasts surge and observed hourly sea level after removal of the tide (and filtering to suppress variations with periods shorter than 12 h that are not resolved by the model). Overall the surge hindcast is quite accurate; the hindcast rms error for the period 1960–1999 is typically 8 cm. The focus of the present study is not the hindcasting of particular storm surges; it is the estimation of flooding probabilities at particular times of the year. Our approach to this problem closely follows that of Bernier and Thompson (2006) but includes an extension to seasonal returns (rather than annual returns). The steps in the calculation of seasonal return periods for a location with, say, only 5 yr of data is as follows: Step 1: The tides, ηT, are predicted for the first 4 yr using a tidal package (e.g., Pawlowicz et al., 2002) and tidal constants fitted to the fifth year of observations. Step 2: The observed sea levels, for the 4 yr not used for the tidal analysis, are written in the form g ¼ gT þ gS þ g R V 143 synthetic ηR′ record obtained by randomly sampling with replacement from the binned ηR′ record. To allow for sampling variability, ten realizations are generated each 40 seasons in length. Step 5: The median of the seasonal maxima of the 10 realizations is used in the standard extremal analysis. The extremal analysis of the reconstructed seasonal maxima is shown in Fig. 4 for Shediac (Fig. 1, station 14) for Winter and Spring. There is an offset between the stormy season of Winter and the quieter season of Spring and a difference in the slope of the two lines. Both are primarily associated with weaker cyclone activity during the Spring season (e.g., Koutitonsky and Budgen, 1991). The ability of our approach to estimate seasonal return periods at all locations is shown in the left column of Fig. 5. The 40 yr return levels for Spring based on the reconstructed records are plotted for the 24 gauge locations. Results show good agreement between the observed and reconstructed seasonal sea levels (compare top and bottom left panels). The 40 yr return levels for Spring based on the reconstructed records are within about 10 cm of the observed Spring return levels. The 40 yr reconstructed levels for Spring also show the same station to station variability as the 40 yr observed return levels for Spring. Thus we can ð3Þ where ηS is the hindcast surge and ηR′ is a hindcast error that includes the effect of seiches and baroclinic effects not captured by the storm surge model as well as errors in predicting the tide based on data from another year. Step 3: The ηR′ are binned by time of year to reflect seasonality in the distribution of ηR′. We used two bins in this study: Spring–Summer and Fall–Winter. Step 4: We predict the tides for 40 yr using the constants previously fitted to the fifth year of observations. (Note that the tidal package of Pawlowicz et al., 2002 does have the option of allowing for the 18.6 yr nodal modulation. We have verified that it is recovered in the predicted tides). We then add, for the season of interest, the 40 yr predicted record of tides, surge hindcast, and a Fig. 4. Seasonal return period of extreme total sea levels at Shediac calculated from the reconstructed seasonal records. The x-axis is the return period of the seasonal maxima in years where xi = − log(− log (1 − pi)) for the ith ordered and adjusted seasonal maximum. The yaxis is the seasonal return level in meters. Note, both the offset and the change in the slope of the return level line between the stormier Winter period (January–March) compared to the quieter Spring period (April–June). 144 N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 Fig. 5. 40 yr return levels. The x-axis shows the station code. The y-axis is the 40 yr return level for Spring in meters calculated from the Type I distribution fitted to the adjusted Spring seasonal maxima (Fig. 4). The shaded areas mark the 95% confidence intervals. The top left panel shows the observed 40 yr return levels. The bottom left panel shows the 40 yr Spring return level obtained using 5 yr of hourly sea level observations and the 40 surge hindcast (Section 2.2). The top right panel shows the 40 yr Spring return level of the filtered residuals with periods below 12 h removed (LηR). The bottom panel is the 40 yr Spring return level of the hindcast surges (ηS). generate seasonal extremes with only 5 yr records and our 40 yr surge hindcast. 3. Downscaling the return periods In this section, a DEM for an important piping plover nesting ground (top panel of Fig. 6) is used to downscale the return period of extreme sea levels for the nesting season (Spring). The approach is to flood the DEM to a specified level (ηc), or equivalently return period, to identify areas at risk. The DEM can therefore be used to evaluate the extent of any hindcast or forecast flood event. One subtlety is that flooding is path dependent e.g., a small valley can only be flooded if the land separating it from the adjacent ocean is also flooded (hence, there must be a path to the valley). Another subtlety is that allowance had to be made for culverts that can affect the extent of flooding but are not detected by the lidar system that was used to make the DEM (Webster et al., 2004). We also note that the DEM is flooded naively in the sense that no delay is allowed for a flood to propagate through a culvert or to retire from a flooded region. Models are available for this purpose and could be used in future studies. The flooding maps are obtained by first performing an extremal analysis on the Shediac reconstructed Winter and Spring record (Fig. 4). The sea level at which each point in the DEM becomes flooded is then associated with a return period. The result is a map of the return period of flood events for the Winter and Spring seasons (middle and bottom panels of Fig. 6 respectively). Fig. 6 shows the different extents of the return period of flooding for the Winter (top panel) and Spring (bottom panel) months. Note how the vulnerable area differs between seasons. Note also how slowly the spatial extent of the area at risk grows with increasing return period in the Spring compared to the Winter. In fact, the 100 yr Spring return level barely exceeds the 5 yr Winter return level. This is the first time such maps have been produced. They have the advantage of simplicity and show, at a glance, which areas are most vulnerable to flooding. Note that flooding is now defined in term of inundated areas (hence landmarks) as opposed to elevation above some arbitrarily chosen datum such as mean sea level. The maps are therefore easy to interpret. Sea level rise is expected to contribute, over the next 25 yr, about 0.2 m sea level increase at Shediac (Section N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 145 Fig. 6. Digital Elevation Model for Shediac, New Brunswick, and spatial maps of the seasonal return period of extreme events. The top panel is the DEM for an area located near station 14 (Fig. 1). The colorbar of the top panel indicates elevation above the DEM datum which is located 21 cm below mean sea level. The horizontal resolution is 1 m and the measured vertical accuracy is better than 0.16 m (Timothy Webster, personal communication). The middle panel is a return period map of Winter extreme events. The return periods are based on the extremal analysis of the reconstructed Winter maxima (Fig. 4). The bottom panel is a return period map of the reconstructed Spring extreme events. 4). On the return period plot (Fig. 4), this rise in sea level will raise the current Spring return level line to roughly the current Winter line. It is therefore expected that the risk of flooding will increase during the nesting period to such an extent that in 25 yr the Spring return period plot and vulnerability map will resemble the current Winter return period plot and vulnerability map (middle panel of Fig. 6). 4. Impact of climate change on the return period of extremes This section briefly explores how climate change may modify the return period of extreme sea levels. The return levels presented thus far are based on a hindcast for a fixed observation period. It is possible to extend our approach to allow for some elements of climate 146 N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 change. For example, it is straightforward to approximate the effect of sea level rise by simply adding a constant height to the return levels as was done in the previous section. However, in the region studied here, it is not possible to add a spatially uniform rate of sea level rise as the effect of vertical crustal movement is spatially variable (Peltier, 2004) and of the same order as the generally accepted rate of global sea level rise of 1– 2 mm yr− 1 (Houghton et al. (2001)). Rates of sea level rise must therefore be station dependent. At Shediac, close to the location of the DEM, relative sea level is expected to have risen 0.8 m by 2100 whereas it is expected to increase by 0.7 m at Halifax. In terms of changes in storminess, the consensus seems to point toward an increase in the number of strong wind events associated with an increase in the number of deep lows, although the overall number of low pressure events is expected to decrease (e.g., Lambert, 2004). The impact of sea level rise and increased storminess on the return period of extreme sea levels are investigated next. Three climate change scenarios are discussed below. They are illustrative rather than definitive and focus on annual rather than seasonal extremes. We also focus on Halifax, an important east coast Canadian city with a long sea level record. The extension of the method to other locations, and from annual to seasonal extremes, is straightforward. For the purpose of this exercise, it is assumed that the low-pass filtered residuals (see Appendix A) recorded at Halifax between 1960 and 1999, represent the current conditions and are denoted by gpres ¼ Lðg−gT Þ ð4Þ where η is the observed sea level about the mean and ηT is the predicted tide. Hence, the observed sea level record at Halifax is given exactly by g ¼ gT þ gpres þ gR V ð5Þ where ηR′ includes, as before, the effects of seiches and baroclinicity. It is further assumed that the impact of climate change can be expressed simply through a mathematical transformation of ηpres. The transformed ηpres will be referred to as ηclim, the climate modified residuals. Scenario 1: It is assumed that atmospheric conditions, i.e., winds and pressure forcing, remain unchanged as we progress into the next century. It is further assumed that sea level at Halifax, the representative station, will have risen 0.7 m by 2100. This sea level rise is imposed on the present observed residuals ηpres to obtain ηclim, the climate modified residuals. The effect on the residuals is illustrated in the left panel of Fig. 7. The black line shows ηpres plotted against itself as a reference. The blue line shows ηclim = 0.7 + ηpres i.e. the present residuals plus sea level rise plotted against ηpres. In terms of the probability density function (or similarly the histogram), applying sea level rise shifts the probability density function of ηclim to the right by 0.7 m (not shown). The shift in the probability density function increases the annual maxima by 0.7 m (and also an, the location parameter of the Type I distribution fit to the annual maxima). This is illustrated in the Type I plot (right panel of Fig. 7). The plot is produced by fitting a line through the ordered annual maxima of η = ηT + ηpres + ηR′ (black line) and η = ηT + ηclim + ηR′ (blue line). The return period plot shows that the impact of sea level rise is so important that extreme sea levels with a current return period of 100 yr are expected to have become regular events by 2100. Scenario 2: It is assumed that in combination with sea level rise, increased storminess will lead to a linear increase in the climate modified residuals. Hence, ηclim = 0.7 + αηpres where α is a scaling factor (α = 1.3 was chosen). The resulting climate residuals are plotted against ηpres (in green) in the left panel of Fig. 7. The linear increase in surge intensity has shifted and stretched the probability density function of ηpres. The location parameter, an is increased via the inflation of the annual maxima. The shape parameter bn, a measure of the spread, is also increased. The return period plot of the ordered annual maxima of η = ηT + ηclim + ηR′ is shown in green in the right panel of Fig. 7. The impact of the linear increase in surge amplitude on the return period plot is two-fold: the return level line is lifted and the slope of the return level line is accentuated. The overall effect is to further reduce the return periods of extreme total sea levels. Scenario 3: It is assumed that in combination with sea level rise, weak cyclones will have a tendency to deepen more while intense cyclones will remain relatively unchanged. The effect on the amplitude of surges is assumed to be as follows: mid-amplitude residuals are inflated while the largest residuals are left untouched. This is illustrated in the left panel of Fig. 7 (red line). The effect of the chosen triangular scaling is to fatten the probability density function at the mid-range amplitudes (i.e. in the positive tail of the distribution) while keeping the range fixed. The impact of increasing mid-amplitude surges appears more important than that of a linear increase in surge amplitude (compare the red and green lines of Fig. 7). This result confirms that it is rarely the most N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 147 Fig. 7. Return period of extreme total sea levels into the next century for Halifax. The left panel shows the climate modified residuals, ηclim, against the present residuals, ηpres. ηpres is plotted against itself in black as a reference. Residuals resulting from sea level rise (blue line) and the combined effect of sea level rise and changes in storminess (green and red lines) are also plotted. The right panel is the return period plot of extreme total sea levels associated with each climate change scenario. Each line is based on an extremal analysis of the annual maxima of each climate modified total sea levels, η = ηT + ηclim + ηR′. The return levels shown are for (i) current conditions (black line), (ii) sea level rise (blue line), (iii) sea level rise and linear increase of residuals amplitude (green line), and (iv) sea level rise and inflation of mid-range surge amplitude (red line). extreme residuals that lead to annual maxima; extreme sea levels are usually the result of a large (but not unusual) residual coinciding with high tide. The results shown in Fig. 7 suggest that over the next century, the continued increase in flooding risk will primarily be brought about by sea level rise (assuming that the increase in sea level is comparable to what is expected for Halifax). The simple sensitivity study presented here also shows that changes in storminess should not be overlooked when forecasting flooding risk into the next century. 5. Conclusions For locations lacking long sea level records, a storm surge model can be used in combination with a short observation record to calculate the return period of extreme events on an annual or seasonal basis. Type I extremal analysis of reconstructed seasonal sea levels in the Northwest Atlantic gave good estimates of the 40 yr return level. The method was also adapted to calculate the return period of extreme events over a season of interest. An important point to note is that this approach requires only a few years of hourly sea level observations to calculate multi-decadal annual and seasonal return levels of total sea level. Given a Digital Elevation Model, it is possible to downscale the results of an extremal analysis to produce maps of the return period of flood extent. The extremal analysis may be performed on the annual or seasonal maxima of either a long observation record or a reconstructed record. The maps can therefore be produced for different periods of the year. Such maps can then be used, for example, as an indicator of risk to bird populations during their nesting season. Climate change effects such as sea level rise and changes in storminess can also be accounted for and maps of the extent of floods may be produced under various climate change scenarios. The maps can then be used to evaluate how sea level rise may disturb the nesting season of species at risk or may damage tourist infrastructure. Thus, DEMs are a powerful way of downscaling the results of extremal analyses to the community level. The return period maps presented in this paper have the advantage of simplicity. They are easy to understand and allow easy and rapid identification of areas most at risk of flooding. They can therefore be used by policy makers to identify zones where development should be limited, adaptation measures should be put in place, or ecosystem management should be considered. Although it has not been discussed in this paper, we have successfully reproduced the return period of 148 N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 minimum sea levels (Bernier, 2005). In areas where maritime traffic requires a minimum water depth for ships to enter ports or navigate safely, maps of the return period and extent of low water levels can also easily be produced. Allowing for changes in the severity of storms is difficult and ultimately such changes should be based on reliable high resolution forcing from global climate models. These are currently not available. We therefore performed preliminary sensitivity studies by modifying the shape of the distribution of observed surges. Changes in storminess represented by large increases in the amplitude of large or mid-range residuals were found to have less impact on the return period of extreme sea levels than the realistic sea level rise predicted for some areas of the study region (e.g., Scotian Shelf ) over the next century. This does not imply that changes in storm severity will have little impact on the return period of extreme events. This sensitivity study rather suggests that in regions where sea level rise is considerable (order 1 m) it alone will result in a dramatic reduction in the return period of extreme events over the next century. We have assumed throughout this study that tidesurge interaction is negligible in this region. This is consistent with the generally good estimates of the extreme seasonal return levels for most tide gauge locations. Nonetheless, including tidal forcing in the open boundary condition of the surge model, and thus allowing for tide surge interactions, could improve the hindcasts of total sea level in some areas. Preliminary results suggest that tide surge interactions may be significant in the southern Gulf of St. Lawrence. This is the subject of an ongoing study. Acknowledgements This work was funded by the Natural Sciences and Engineering Research Council of Canada and the Climate Change Action Fund. The Digital Elevation Model was kindly provided by Tim Webster of the Center of Geographic Sciences, Nova Scotia. Tim Webster also inserted the culverts (see text) and was generous with the provision of advice on processing of the Digital Elevation Model and also additional information on its accuracy. Finally, we would like to thank the reviewers for their constructive comments. Appendix A. 40 yr surge hindcast The surge hindcast model is a modified version of the Princeton Ocean Model (Blumberg and Mellor, 1987). It is based on the following depth averaged barotropic momentum and continuity equations Au t s −t b þ udju þ f u ¼ −gjgW þ þ Aj2 u At qh ðA:1Þ AgW AðHuÞ AðHvÞ Ag þ ¼− P þ Ax Ay At At ðA:2Þ where u = (u, v) is the depth-averaged horizontal velocity, f is the upward pointing unit vector scaled by the Coriolis parameter, τs and τb are the surface and bottom stress, h is the mean water depth, A is the horizontal viscosity, and H is the total water depth. The rest of the notation is standard. The model's sea level (ηS, where the subscript s denotes surge) has been written in the form (e.g. Gill, 1982) gS ¼ gP þ gW ðA:3Þ where ηP is the barometer effect due to variations in air pressure and ηW is the isostatically-adjusted sea level of the model. In the model, ηW is formulated as the prognostic variable (Bobanović et al., 1997). ηW is due to the effect of the wind and the dynamic response of sea level to air pressure forcing. The kinematic stress magnitude, cd (W)W2, is given by 1:2 10−3 ; jW jb8 ms−1 cd ¼ 0:68 10−3 þ 0:065 10−3 u; jW jz8 ms−1 ðA:4Þ where W is the wind speed. The bottom stress formulation is of the form cdbu(u2 + v2)1/2 where cdb, the bottom drag coefficient, equals 2.5 × 10− 3. The model domain extends from 38° N to 60° N and 72° W to 42° W (Fig. 1). It covers the Labrador and Newfoundland Shelves, the Gulf of St. Lawrence, the Scotian Shelf, the Bay of Fundy and the Gulf of Maine. The surge model resolution is 1/12° which corresponds to a latitudinal resolution of about 9 km. The surge model was forced with AES40, a set of high resolution, 6 hourly reanalyzed winds (Swail and Cox, 2000; Swail et al., 2000) and with AESP40, the pressure fields inferred from the AES40 surface wind fields (Bernier and Thompson, 2006). The result is a 40 yr hindcast (1960–1999) of hourly sea levels. Fig. 3 is an example of the model output. It shows the propagation of a large surge event. The surge hindcast model is driven by winds and pressures provided every 6 h. This corresponds to a Nyquist frequency of one cycle every 12 h (Priestly, 1981). N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 Fig. A.1. Root mean square hindcast error in meters. The x-axis is station code (see Fig. 1 for station location). The y-axis is the rms hindcast error. Thus, apart from the contribution of nonlinear effects, the surge model does not generate variability at periods shorter than 12 h. To compare the hindcast surges and the observed residuals we removed variability in the residuals at periods shorter than 12 h using a low-pass filter LgR ¼ Lðg−gT Þ ðA:5Þ where LηR is the low-pass filtered residual. The hindcast surges were compared against the observed residuals at 149 the 24 available gauges (see Fig. 1 for station locations). The rms error over the 40 yr of the hindcast was typically 8 cm (Fig. A.1). Rms errors calculated on 1 yr increments confirmed that the quality of the hindcast remained comparable throughout the 40 yr of the hindcast (Bernier, 2005; Bernier and Thompson, 2006). Another way to validate the hindcast is to use the γ2 statistic, a measure of the variance of the hindcast error upon the variance of the observed residuals (Thompson et al., 2003). This form of statistic is useful in determining whether the hindcast captures the general evolution of the residuals. Fig. A.2 illustrates how γ2 varies spatially (each γ2 is calculated using the selected observation record and prediction at different model grid points). As expected, the skill of the model worsens as we move away from the observation site. This type of plot can be useful in identifying problems in areas where the coastline surrounding a gauge is rugged, a gauge is located in a bay not well represented by the model resolution, or where two or more grid points could justifiably be chosen to represent a tide gauge location. Fig. A.2. Map of the variance of the hindcast error upon the variance of the filtered observed residuals (γ2) at Rivière-au-Renard. The colorbar indicates γ2. The location of the LηR record used to measure the γ2 is indicated by the yellow dot. γ2 typically varies slowly at grid points adjacent to the gauge location. It then increases with increasing distance from the gauge. When γ2 reaches a value of 1, the variance of the hindcast error is as large as the variance of the signal. 150 N.B. Bernier et al. / Global and Planetary Change 57 (2007) 139–150 The γ2 statistic does not tell us how well any given large surge event is reproduced. The return period of surges were therefore compared to return periods of LηR (Section 2.2). The agreement is good with the hindcast return levels typically within 10 cm of the observed filtered residuals for all seasons and for both maxima and minima (Bernier, 2005), thus confirming the quality of the hindcast. References Bernier, N.B., Thompson, K.R., 2006. Predicting the frequency of storm surges and extreme sea levels in the northwest Atlantic. Journal of Geophysical Research-Oceans 111 (C10009). doi:10.1029/2005JC003168. 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