A methodology to define extreme wave climate using reanalysis data bases

Transcription

1 A methodology to define extreme wave climate using reanalysis data bases F. J. Méndez,A.Tomás,R.Mínguez, and B. G. Reguero Environmental Hydraulics Institute IH Cantabria, Universidad de Cantabria Avenida de los Castros s/n, Santander, Spain Abstract Hindcast or Wave Reanalysis Data Bases (WRDB) have become a powerful tool for the design of offshore and coastal structures, since they offer important advantages for the statistical characterization of extreme events (continuous time series, good spatial coverage, constant time span, homogeneous forcing, > 40 year long time series). However, WRDB are affected by several limitations that must be addressed prior to their use (they are not quantitatively perfect; high-frequency information is not included in global wind reanalysis models). In addition, the selection of a given extreme value model is also an important issue that must be conveniently analyzed. The objective of this work is to describe a complete methodology able to cope with the problems derived from the use of WRDB, specially for extreme value analysis and long term design. We pay special emphasis to: i) the high frequency noise or peaks of the time series, and ii) the extreme value statistical model. The methodology is illustrated with examples applied to the port of Barcelona (Spain). I. INTRODUCTION AND MOTIVATION Coastal and offshore structures are subject to a life cycle process encompassing different phases: i) planning and design, ii) construction, iii) operation and maintenance, and iv) re-used and/or dismantling. During each of these phases, the structure and the environment undergo a continuous sequence of outcomes, the consequences of which have to be considered in the project. The objective of the design is to verify that the structure satisfies the project requirements during these phases in terms of acceptable failure rates and cost, achieving cost effective functionality (f.e. serviceability, availability, etc.) and technical quality (f.i. safety, environmental, durability, sustainability, etc.). Generally, marine structures are designed for a minimum life of years, and satisfaction of the safety requirements mostly depends on marine climate (sea level, waves, astronomical tides, storm surges, wind and currents) on the long term trend. In the last years, the development of wave reanalysis models allow a detailed description of wave climate in locations where long-term buoy records are not available. For this reason they have become a powerful tool used for the design of offshore and coastal structures. However, reanalysis models are a simplification of reality which also use discrete forcing fields consisting of surface winds at different times, and quantitative results present differences when comparing with recent instrumental data (buoys and/or satellite) (see [1] and [2]). This paper presents the guidelines to use WRDB for the long-term design of maritime structures, which is able to cope with the problems and imprecisions of this kind of data. We pay special emphasis to i) calibration using instrumental data, ii) inclusion of high frequency noise and iii) selection of the appropriate extreme value statistical model. This allows a more accurate description of the extreme value analysis, deriving in cheaper and/or safer designs. The different aspects of the methodology are illustrated with examples applied to the port of Barcelona (Spain). II. METHODOLOGY Long-term design of maritime structures requires the knowledge about the maximum forces the design must withstand during the lifetime of the /11/$ IEEE

2 Reanalysis wave data base Wave climate shallow waters Satellite and deep water buoy data Fig. 1. Directional calibration High frequency correction Shallow water buoy data Selection Bathymetry and sea level data Propagation Extreme value analysis and design Extreme model Methodology flow chart. structure. These maximum forces are mostly related to maximum significant wave heights, and therefore, an accurate maximum wave height analysis in the location of interest is required. Extreme value analysis concerns the knowledge about the occurrence of extreme events and their frequency, and a careful analysis requires the availability of data on such extremes. The larger the size of the data record, the more accurate the statistical model for those extremes will be, which would lead to better predictions. Since in many locations there no exist long data records to be used for extreme value analysis, wave reanalysis databases seems to be a plausible alternative. However, in order to use this kind of data appropriately, the following steps must be covered (see flowchart in Figure 1: 1) Selection of the wave reanalysis data base: Wave climate information from a wave reanalysis database must be selected. These data characterize deep waters wave climate around the location of interest, as shown in Figure 1. 2) Directional calibration: Since wave reanalysis climate in deep waters is not quantitatively perfect, different corrections depending on the wave direction are performed. Note that the calibration procedure uses instrumental data (satellite and buoys). 3) Case selection: At this point, a very long record, which may be up to several hundred thousands elements, containing the information of the sea state on an hourly basis is available. The information includes significant wave height (H s ), mean period (T m ), peak period (T p ), wave direction (θ m ), and wind velocities (W x,w y ). This information can be used to propagate to shallow waters by nesting a numerical model which simulates the wave transformation processes in shallow waters. However, the propagation of all possible sea states from the reanalysis node to the location of interest is computationally infeasible. For this reason, a prior selection of the cases to be propagated which are representative of all population is performed [3]. 4) Case propagation: Each selected sea state case is propagated using the last version of the SWAN model [4]. The characteristics of the propagated sea states are obtained at the location of interest. 5) Shallow waters wave climate reconstruction: The time series of the propagated parameters (H p s,t p m,t p p,θ p m) for all deep water sea states are reconstructed by a multilinear interpolation technique based on radial basis functions (RBF). The approximation function of each parameter is formed by a lineal combination of radial basic functions centered in the scattered points defined by the selected cases, with the associated real function values which are the corresponding propagated parameters. In this work a Gaussian function has been used in these interpolation technique and the optimal shape parameter has been obtained by [5] based on leave-one-out cross validation. 6) High frequency correction: Due to the temporal resolution of global wind reanalysis models (i.e. NCEP/NCAR, ERA-40 or JRA), which are the forcing for wave reanalysis models. The physical processes associated with the high frequency bands are not appropriately reproduced. This is especially relevant for the peaks of the time series, i.e. maximum significant wave heights. This motivates an additional correction using high frequency information from shallow water buoys. 7) Extreme value analysis: Once the wave climate time series in the location of interest is available, the extreme value analysis is

3 performed. This allows to calculate significant wave heights associated with the design return periods. For this task, we propose to use the last state of the art non-stationary extreme value model based on monthly maxima. This model allows accounting for seasonality in the design process, and it increases the confidence in the predictions. In the following sections, we explain the complete methodology in more detail, paying especial emphasis on steps 2) calibration, 6) high frequency correction and 7) extreme value analysis. A. Selection of the wave reanalysis data base Wave hindcasting usually refers to a numerical model integration over a historical period without assimilating observations, since oceanographic observations such as the significant wave height are much scarcer than meteorological observations, and it has been considered adequate for generating a reasonable representation of the wave climate with little need for a full reanalysis. However, reanalysis models incorporate observational information within the process. We have used the following databases: 1) SIMAR-44 generated by Puertos del Estado. They used the 44-year ( ) dynamic downscaling REMO [6] from the global atmospheric re-analysis carried out by the National Centre for Environmental Prediction, Washington, USA (NCEP) and the National Centre for Atmospheric Research, Boulder, Colorado, USA (NCAR) and the wave models WAM [7]. This SIMAR-44 reanalysis consist on hourly time series over a 44- year period ( ) of significant wave height (H s ), mean period (T m ) and mean direction (θ m ) over different regular grids around Spain. 2) GOW database (Global Ocean Waves) generated by IH Cantabria using WaveWatch III model [8], [9]. This database provide spectral sea state parameters: significant wave height (H s ), mean period (T m ), peak period (T p ) and mean direction (θ m ) as well as the directional spectra components S(f,θ), along the coast. There are two versions: a) GOW 1.0 (Global Ocean Waves 1.0). Hourly reanalysis for the period between , with global coverage and along the Spanish coasts. Atmospheric forcing taken from the NCEP/NCAR reanalysis validated by satellite data by IH Cantabria. b) GOW 2.1 (Global Ocean Waves 2.1). Hourly reanalysis for the period between with resolution in the Mediterranean and Cádiz Gulf. Atmospheric forcing taken from the dynamic regional downscaling SeaWind, generated by the IH Cantabria, with 15 km spatial resolution, nested with the atmospheric reanalysis ERA-Interim. In the West boundary the hourly directional spectra taken from the GOW 1.0 reanalysis are introduced. B. Directional calibration Due to the characteristics of reanalysis models, which are primarily feeded using wind data, it is known that inaccuracies of WRDB are mostly dependent on the bad description of the wind fields, i.e. insufficient forcing resolution. The quantitative differences between numerical and instrumental data suggests that different corrections should be applied depending on the mean direction of the sea state, i.e. for directions where the wind resolution is not enough to capture the local wind wave generation, but not for swell waves generated in areas where the wind resolution is sufficient to reproduce the wave dynamics. In this paper we propose the use of a new parametric calibration method based on a nonlinear regression problem with the following characteristics: i) it manages to combine buoy, satellite and model data, ii) the correction parameters vary smoothly along the possible mean wave directions by means of cubic splines, allowing different corrections depending on the wave direction, iii) corrections are made on empirical quantile information on a Gumbel probability paper scale giving more weight on the calibration procedure to the maximum data, which is more important from the design point of

4 Location of interest Propagation processes (Refraction + shoaling + difraction + breaking ) + Local wind generation Wave climate in shallow waters Reanalysis node Wave climate in deep waters (WRDB) Wave climate in deep waters (WRDB) kh>π Fig. 2. Deep and shallow water wave records, location of interest and wave reanalysis node. view. The nonlinear regression model is solved using the least squares (LS) method, where we minimize the sum of squared distances between observed and predicted values, that is, n ( Minimize ε T ε = a R,b R i=1 where ε are the residuals, which are assumed to be uncorrelated and identically distributed normal random variables with zero mean and unknown constant variance, n is the number of observations, corresponds to buoy and satellite data, and I R is the reanalysis significant wave height. and a R (θ) and b R (θ) are the parameters dependent on the wave direction θ. Note that although we particularize equations for significant wave height variables, the method is also valid for other reanalysis variables such as mean wave periods. Note that C = a R (θ) [ ] R b R (θ) is the calibrated significant wave height The model relies on the assumption that parameters a R and b R vary smoothly with the propagation direction (θ). These variations are introduced in the model throughout cubic splines, so that only a given number n p of values of the parameters at different given directions a j,b j ; j =1,...,n p are known. The parameter values for all possible directions are obtained by interpolation using smoothing cubic spline functions: I i a R (θ i ) [ a R ] 2 b R (θ i)) R i (θ i ) = a j + x a j (θ i θ j )+yj a (θ i θ j ) 2 i,(1) +zj a (θ i θ j ) 3, (2) b R i (θ i ) = b j + x b j(θ i θ j )+yj(θ b i θ j ) 2 +z b j(θ i θ j ) 3, (3) where a R i and b R i are the interpolated model correction parameters for a given direction θ i, a j,b j ; j = 1,...,n p are the parameters to be estimated, i.e. the parameter values associated with directions θ j ; j = 1,...,n d, and x a j, ya j, za j, xb j, yb j, z b j ; j = 1,...,n d are the corresponding cubic spline parameters, which are obtained using zero, first and second order continuity conditions along the circumference (0 θ 2π). A detailed description of the methodology can be found in [10]. C. High frequency correction A visual inspection of the spectra of the Significant Wave Height (SWH) instrumental time series I, versus calibrated WRDB time series C (see

5 ~= 0.4 m ZOOM H I (t) s H C (t) s I (t) H LF (t) Fig. 3. Spectra of the instrumental and calibrated significant wave height (SWH) time series (left panel) and instrumental SWH versus its SWH associated with the low frequency (1 h 1 ) energy (right panel) for Villano buoy (Coruña). left panel in figure 3) reveals that there are important differences in the 1 h 1 to 1/6 h 1 frequency band. The main reason for this gap is the temporal resolution of the global wind reanalysis model (i.e. NCEP/NCAR, ERA-40 or JRA) which provides wind velocities every 6 hours, and besides spatial resolution is not enough to model the physical processes which affect high frequency energy, for instance currents. We have applied a Fast Fourier Transform (FFT) 6-hour filter to smooth the instrumental (H I s ) time series, which can be decomposed as: H I s = H HF s + H LF s, (4) where HF and LF are the SWH associated with high and low frequency energies respectively. Comparing the calibrated WRDB time series C with respect to the low frequency time series LF, it can be shown that both signals present a similar spectral behavior. This result confirms that the WRDB is unable to reproduce the wave height HF related to high frequency band. This high frequency noise is especially relevant for the peaks of the time series, as shown in the lower panel of figure 3, where s there are differences up to 0.4 meters. This aspect is determinant for a correct definition of the extreme events and it must be included in the wave record. The scatter plot calibrated WRDB time series C versus high frequency SWH HF (see figure 4) reveals that the latter conditioned to a given value of C follows a normal distribution with parameters μ H C s and σ H C s. For this reason we propose the following parameterization of the conditional mean and standard deviation: μ H C s = a μ + b μ C, (5) σ H C s = a σ + b σ C, (6) assuming that both the mean and standard deviation vary linearly with respect to the calibrated SWH, C. Parameters a μ, b μ, a σ, and b σ are estimated using the method of maximum likelihood. Once the parameters of the conditional distribution are estimated, the next step is to make the correction to include the high frequency noise. For this task we propose solving the following optimization problem: Minimize q MSE = n i=1 ( H HF s i e i ) 2, (7) where q correspond to a given probability (0 q 1), and e i is the additional correction which must be applied to the calibrated WRDB time series C so that the final shallow water wave climate is obtained as F i = C i + e i. This correction e i is equal to e i = μ H C si Φ 1 (q)σ H C si. The optimal solution q of problem (7) corresponds to the probability associated with the quantile of the fitted noise distribution HF i which makes the least square differences (MSE) between observed noise and corrections to be minimum. Note that the correction is equal to the corresponding conditional quantile. This approach is based on the following assumptions: 1) Since we are interested in maximum values, the use of least squares automatically give more weight to the larger significant wave heights. 2) We consider the worst case scenario, and for this reason and although the high frequency

6 H HF s H C s = 2.9 H HF s H C s = 3.0 H HF (t) s H HF s H C s = 3.1 H HF s H C s = 3.2 H C (t) s Fig. 4. Scatter plot calibrated WRDB time series C versus high frequency SWH HF functions and histograms for different C -values for Barcelona port f4f4f4data. and conditional probability density noise may be positive or negative we always add a given quantile belonging to the right tail of the distribution (q >0.5). This assumption is conservative. D. Extreme value analysis Recent advances in the extreme value theory [11], [12] allow modeling the natural climate variability of extreme events of ocean climate variables, such as wave height. These methods introduce timedependent variations within a certain time scale (year, season or month), improving our knowledge on some important natural coastal processes relevant for engineering design. Among the approaches considering seasonal variability, [13] propose a time-dependent model based on the GEV distribution that accounts for seasonality, using independent monthly maxima events observed at different instants, thus considering 12 maximum values per year, is developed. This approach results in a reduction of the uncertainty in the estimation of time-dependent (monthly) quantiles, and the improvement in the estimation of annual return values. This return values are crucial for design purposes. Within their approach, monthly maxima of successive months are assumed to be independent random variables, but the hypothesis of homogeneity through consecutive months is not needed (because they are not presumed to be identically distributed). Monthly maximum x t of the climate variable observed in month t follows a GEV distribution with time-dependent location parameter μ t, scale parameter ψ t, and shape parameter ξ t, with a cumulative distribution function (CDF) given by: [ ( )] 1 exp xt μ t ξ 1+ξ t t ψ t + ; G(x t )= (8) { [ ( )]} ξ t =0, xt μ t exp exp ; ξ t =0, where [a] + =max(0,a), and the support is x t μ t ψ t /ξ t,ifξ t < 0 (Weibull), or x t μ t ψ t /ξ t,ifξ t > 0 (Fréchet). The corresponding time dependent quantiles x q,t are: x q,t = ψ t [ ] μ t ψt ξ t 1 ( log q) ξ t, if ξt =0, (9) μ t ψ t log( log q), if ξ t =0.

7 Annual quantiles x q corresponding to periods equal to or longer than one month, given by the interval (t a,t b ), can be obtained solving the { following implicit equation: tb [ ( )] 1 } xq ξ μ t t exp k m 1+ξ t dt ; t a ψ t + q = { ξ t =0, tb [ ( )] } (10) xq μ t exp k m exp dt ; t a ψ t ξ t =0, where 1/k m =1/12 year. The GEV distribution includes three distribution families corresponding to the different types of tail behavior: Gumbel (null shape parameter); Fréchet distribution (ξ t > 0); and Weibull family (ξ t < 0). To introduce seasonality, the model proposed in [13] is as follows: i=1 P ξ ξ t = γ 0 + [γ 2i 1 cos(iwt)+γ 2i sin(iwt)], i=1 where t is given in years, log (ψ t ) ensures positiveness of the scale parameter (ψ t > 0), β 0, α 0, and γ 0 are mean values, β i, α i, and γ i are the amplitudes of harmonics considered in the model, w =2π/T is the angular frequency, T is one year, and P μ, P ψ and P ξ are the number of sinusoidal harmonics to be considered within the year, associated with the location, scale and shape parameters, respectively. A particular model is represented by the following parameter vector of dimension n p : θ =(β 0,β i,α 0,α i,γ 0,γ i ), (11) where given n d observations of monthly maxima x = (x 1,x 2,...,x nd ) T occurring at times t = (t 1,t 2,...,t nd ) T. Model parameters are estimated using the method of maximum likelihood. Issues related to the automatic fitting and sensitivity analysis of the method can be found in [14], [15]. Fig. 6. Llobregat Buoy Location of the study area. Note that this method requires, using the final shallow water wave climate time series F,the selection of maximum significant wave heights for every month. In order to hold the independency assumption, maximum data can not be located on the same 6 days time moving window, i.e. they can not belong to the same storm. Figure 5 shows three P μ different goodness of fit plots corresponding to the μ t = β 0 + [β 2i 1 cos(iwt)+β 2i sin(iwt)] H i=1 s F time series related to a given buoy location P ψ close to the Port of Barcelona. As it is shown, the fitting presents very good diagnostics with points in log (ψ t ) = α 0 + [α 2i 1 cos(iwt)+α 2i sin(iwt)] the QQ and PP plots close to the diagonal III. BARCELONA PORT CASE STUDY In order to show the functioning of the proposed methodology in a realistic case study, and the importance of the high frequency correction in extreme value analysis, we have selected several locations in front of the Port of Barcelona, which is located in the North East coast of Spain (see figure 6). Note that one of the selected locations corresponds to the position of a shallow water buoy (Llobregat). This buoy data will be used for validation. The remainder points are positioned over different locations around the port. The proposed methodology has been applied in the seventh locations shown in figure 6: 1) Deep water buoy data from several SIMAR- 44 reanalysis database has been selected. 2) Wave records are calibrated using instrumental and buoy data. 3) Shallow water wave height records at the different selected locations are generated using the selection, propagation and reconstruction methods explained in section II

8 (b) (a) (c) Fig. 5. Goodness of fit plots for the Barcelona Port buoy data location corresponding to the best model: (a) Maximum significant wave height data, location and scale parameters of the best model and 20-year return period quantile (95%), (b) probability plot, and (c) quantile plot. For the final extreme value analysis, we have considered two different situations: 1) Fitting monthly maxima using the nonstationary GEV including the high frequency correction obtained through the Llobregat buoy record. This analysis is performed for locations. 2) Fitting monthly maxima using the nonstationary GEV without considering the high frequency correction. This would allow us to compare the effect of the high frequency energy in the extreme value analysis. Figure 7 shows yearly return periods (red line) obtained using the aggregated quantile expression (10), 90% confidence bands (dashed red lines), the 12 monthly return periods (solid lines), and the plotting positions associated with the data used for the fitting process. Yearly values represent the significant wave height (H s ) which is exceeded on average once every T years, while monthly values represent the significant wave height (H s ) which is exceeded on average once every T januaries, jebruaries, and so on, depending on the corresponding month. Note that the maximum return period plotting position is related to the length of the recor in years, i.e. 44 years (the length of the database). Note that the left panel presents the results including the high frequency correction, while the right panel does not consider the correction. Based on these results, it can be concluded that not including the high frequency information in the extreme value analysis may lead to under prediction of the design wave height. Note that the 44 years return period quantile is about 5 and 4.70 meters, respectively, which represents a 30 cm difference. This may lead to an unsafe design. On the other hand, this quantile value coincides with the maximum significant wave height record from the buoy. This confirms the validity of the model including high frequency correction. Analogous results are obtained for the remainder locations. In figure 8 results associated with location 13 are shown. Note also that the difference for the 44 year return period quantile is also around 30 cm.

9 LLOBREGAT BUOY High frequency effect (approx. 30 cm) Fig. 7. Monthly and yearly return periods for the shallow waters wave climate reconstruction in the Llobregat buoy location considering: a) the high frequency correction and b) no high frequency correction. Location 13 High frequency effect (approx. 30 cm) Fig. 8. Monthly and yearly return periods for the shallow waters wave climate reconstruction in location 13 considering: a) the high frequency correction and b) no high frequency correction.

10 IV. CONCLUSIONS Based on the result obtained using the proposed methodology, the following conclusions can be withdraw: 1) WRDB constitute a plausible alternative for extreme wave climate analysis. Specially for locations where no buoy records or short time records exists. 2) Due to the deficiencies in reanalysis models, mainly provoked by insufficient spatial and temporal forcing resolution, several corrections to reanalysis wave databases must be address. 3) Not including high frequency information by means of the proposed correction may lead to under prediction of the design parameters. The methodology has been successfully applied to a particular case, Barcelona Port. ACKNOWLEDGMENT This work was partly funded by projects GRAC- CIE (CSD , Programa Consolider- Ingenio 2010) and AMVAR (CTM ) from Spanish Ministry MICINN, by project C3E ( ) from the Spanish Ministry MAMRM and by project MARUCA (E17/08) from the Spanish Ministry MF. R. Mínguez is indebted to the Spanish Ministry MICINN for the funding provided within the Ramon y Cajal program. The authors acknowledge to Puertos del Estado the availability of SIMAR-44 database and REDCOS coastal buoy network for this study. REFERENCES [5] S. Rippa, An alkgorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math., vol. 11, pp , [6] D. Jacob, U. Andrae, G. Elgered, C. Fortelius, L. P. Graham, S. D. Jackson, U. Karstens, C. Koepken, R. Lindau, R. Podzun, B. Rockel, F. Rubel, H. B. Sass, R. N. D. Smith, B. J. J. M. Van den Hurk, and X. Yang, A comprehensive model intercomparison study investigating the water budget during the BALTEX-PIDCAP period, Meteorology and Atmospheric Physics, vol. 77, no. 1 4, pp , [7] S. Hasselman, K. Hasselman, P. A. E. M. Janssen, G. T. Komen, L. Bertotti, P. Lionello, A. Guillaume, V. C. Cardone, J. A. Greewood, M. Reistad, L. Zambresky, and J. A. Ewing, The WAM model: a third generation ocean wave prediction model, Journal of Physical Oceanography, vol. 18, no. 12, pp , [8] H. Tolman, User manual and system documentation of WAVEWATCH-III version 1.15, NOAA/ NWS/NCEP/OMB, Technical Note 151, 1997, 97 p. [9], User manual and system documentation of WAVEWATCH-III version 1.18, NOAA/ NWS/NCEP/OMB, Technical Note 166, 1999, 110 p. [10] R. Mínguez, A. Espejo, A. Tomás, F. J. Méndez, and I. J. Losada, Directional calibration of wave reanalysis databases using instrumental data, Journal of Atmospheric and Oceanic Technology, 2011, sent for publication. [11] S. Coles, An introduction to statistical modeling of extreme values. Springer Series in Statistics, [12] R. W. Katz, M. B. Parlange, and P. Naveau, Statistics of extremes in hydrology, Advanced Water Resources, vol. 25, pp , [13] M. Menéndez, F. J. Méndez, C. Izaguirre, and I. J. Losada, The influence of seasonality on estimating return values of significant wave height, Coastal Engineering, vol. 56, no. 3, pp , [14] R. Mínguez, F. J. Méndez, C. Izaguirre, M. Menéndez, and I. J. Losada, Pseudo-optimal parameter selection of nonstationary generalized extreme value models for environmental variables, Environmental Modelling & Software, vol. 25, pp , [15] R. Mínguez, M. Menéndez, F. J. Méndez, and I. J. Losada, Sensitivity analysis of time-dependent generalized extreme value models for ocean climate variables, Advances in Water Resources, vol. 33, pp , 2010, /j.advwatres [1] S. Caires and A. Sterl, A new non-parametric method to correct model data: Application to significant wave height from the ERA-40 reanalysis, Journal of Atmospheric and Oceanic Technology, vol. 22, pp , [2] L. Cavaleri and M. Sclavo, The calibration of wind and wave model data in the mediterranean sea, Coastal Engineering, vol. 53, pp , [3] P.Camus,F.J.Méndez, R. Medina, and A. S. Cofiño, Analysis of clustering and selection algorithm for the study of multivariate wave climate, Coastal Engineering, 2011, in press. [4] N. Booij, R. C. Ris, and L. H. Holthuijsen, A thirdgeneration wave model for coastal regions. Part I: model description and validation, Journal of Geophysical Research, vol. 104, no. C4, pp , 1999.