Prediction of significant wave height using spatial function

Ocean Engineering 106 (2015) 220–226

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Prediction of significant wave height using spatial function Abdusselam Altunkaynak Faculty of Civil Engineering, Hydraulics Division, Istanbul Technical University, Maslak 34469 Istanbul, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 18 August 2014 Accepted 22 June 2015 Available online 28 July 2015

Determining the contribution of variables from stations surrounding a pivot station to variables at the pivot station is very important for many purposes. In this regard, a Regional Dependency Function (RDF) among the variables needs to be obtained. RDF can be used to estimate missing data, determine the location and optimum number of measurement stations (station network design), estimate the potential of a variable under consideration and calculate radius of influence. However, conventional geostatistical methods cannot be employed to achieve the above mentioned uses as they have a number of limitations. As a result, a new method called Slope Point Cumulative Semi-Variogram (SPCSV), was developed to obtain RDF and to address all the limitations of the conventional geostatistical methods. SPCSV was developed by using data from 22 wave measurement stations located off the west coast of the United States. The objective of the study was to predict the significant wave height and determining the influence of radius of the pivot station using this method. Also, the SPCSV method was compared with two other geostatistical methods known as Point Cumulative Semi-Variogram (PCSV) and Trigonometric Point Cumulative Semi-Variogram (TPCSV) using the same data set by taking the mean relative error (MRE) as a performance evaluation criterion. The MRE of the SPCSV method was found to be 6%, which is acceptable in engineering applications. The superiority of the SPCSV method in predicting the significant wave height over the PCSV and TPCSV methods is presented both numerically and graphically. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Prediction Radius of influence Regional dependency function Significant wave height Spatial analysis

1. Introduction Prediction of significant wave height values is an important prerequisite for designing coastal and offshore structures (Soares and Scotto, 2001). According to Bidlot and Holt (1999), wave forecasting has become an integral part of operational weather forecasting at several weather forecasting centers. Saetra and Bidlot (2002) indicated that the incorporation of wave models into numerical weather prediction models can improve atmospheric forecasts by allowing the transfer of momentum between the ocean's surface and the atmosphere to be better modeled. A few techniques have been developed for estimating significant wave height. These techniques range from simple statistical methods to complex polynomial curve fitting models. Roulston et al. (2005) used numerical prediction models to forecast significant wave height probabilities. Altunkaynak and Ozger (2004) developed a technique called Perceptron Kalman Filtering to predict significant wave height from wind speed. Artificial Neural Networks (ANNs) have been successfully used in tracking, retrieval, reconstruction and prediction of waves with the objective of improving the accuracy of numerical models (Makarynskyy, 2005). Paplinska-Swerpel and Paszke (2006) applied the ANN technique to undertake short-term wave forecasts.

E-mail address: [email protected] http://dx.doi.org/10.1016/j.oceaneng.2015.06.028 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

The neural network model was used to predict significant wave height at a selected location in the Baltic Sea based on wave and/or wind data at 10 points scattered over the sea. Makarynskyy and Makarynska (2007) proposed a site-specific ANN methodology that could serve as a basic tool for predicting both present and future wave parameters in various coastal environments. The methodology was recommended as an alternative way of supplementing data and computational effort to demanding deterministic wave models. ANNs have a number of advantages. One of these advantages is that ANNs require less formal statistical training. They have the ability to implicitly detect complex nonlinear relationships between dependent and independent variables and the ability to detect all possible interactions between predictor variables. They are also known to have multiple training algorithms. However, their “black box” nature, greater computational burden, proneness to over fitting and the empirical nature of the models' development are some of the limitations of ANNs (Tu, 1997). Altunkaynak (2008a) developed a method called Geno–Kalman Filtering by combining genetic algorithm and Kalman Filtering method to estimate wave parameters and uses adaptive calculation to reach the solution. In addition, a method called Geno-Multilayer Perceptron was introduced to predict significant wave height by Altunkaynak (2013). The above mentioned techniques, however, were developed to forecast site-specific, short or long term significant wave heights and, therefore, they cannot be directly used for spatial analyses. As a result, surfaces, which can be used as basic information to

A. Altunkaynak / Ocean Engineering 106 (2015) 220–226

perform further spatial analyses need to be generated from available data sources at individual stations. Data required for generating these surfaces are usually collected through field sampling and surveying. Because of the high establishment and operational cost and limitations of other resources, a limited number of stations could be established at selected points and a limited amount of data can be collected as a result. In order to generate a continuous surface of a property (for example: wave energy potential, groundwater table etc…), some kind of interpolation technique has to be used to estimate surface values at those locations where no samples or measurements were taken. The theory of Regionalized variable (ReV) was first developed by Matheron (1963), (1971) to deal with various aspects of a geospatial variable as an alternative technique. The basic principles of regional predictions are based on the study of Krige (1951), which assumed that the spatial variation of any geological, soil or hydrological property, known as a ‘regionalized variable’ is statistically homogenous throughout a surface. This means that the same pattern of variation can be observed at all locations taken into consideration. The Kriging method, which is a weighted moving average based method of interpolation (gridding), was later developed and became the most advanced technique among the regional prediction methods. However, this technique does not have the ability to determine the so-called ‘radius of influence’, which helps to determine the number and spacing of wave power harnessing facilities. This calls for the need to develop a technique that is capable of determining the radius of influence. At the same time, the technique should depict the variation of a variable within a region and should be able to develop surface data by interpolation so that missing data could be estimated. The technique could also be developed so that it can be used for calculating the potential of a variable as well. For example, a spatial interpolation technique can be employed to generate the wave energy potential and its variation at various points within a certain domain by using ocean wave data measured at few representative points. In the presence of spatial relationship (spatial dependency), stations located closer to each other show more similarities than those located far apart. This shows that the similarity between the stations and the contribution of a station to a pivot station decreases with the increase in distance between them. Actually, the presence of a spatial structure where observations at stations close to each other are more alike than those from that are far apart (spatial dependency) is a prerequisite to the application of spatial interpolation (Goovaerts, 1999).

221

As indicated earlier, taking a certain variable into consideration, the same pattern of variation may be observed at all locations on the surface. This spatial variation of a property can be expressed in terms of semi-variograms. A Semi-variogram (a technique proposed by Matheron (1963)) is a graph that shows the variance in measure with distance between all pairs of sampled locations. It is one of the significant functions that indicate spatial correlation in observations taken at sample locations. Despite this advantage, the semi-variogram technique is also known to have some limitations. One of these limitations is the requirement of a uniform distribution of grid points and stationarity in the spatial series. If one does not achieve these requirements, the results will not be valid. In order to deal with the case of irregular distribution of nodal points and non-stationarity, Sen (1989) proposed the Point Cumulative Semi-Variogram (PCSV) method based on point and area relationship. The differences between the point of interest (concern) and the adjacent points are considered in the PCSV technique. These differences are calculated using the variables at each nodal point. Sen and Habib (1998) developed the standard areal dependency approach to make point and spatial estimations of a variable. Later, Altunkaynak and Ozger (2005) used PCSV to predict significant wave height. In the PCSV approach, it is assumed that the highest correlation exists between the pivot station and the closest station to the pivot station, and the lowest correlation exists between the pivot station and the farthest station from the pivot station. Here, it should be noted that this assumption is not necessarily true. The Trigonometric Point Cumulative Semi-Variogram (TPCSV) technique, which was proposed by Sahin and Şen (2004), was applied to deal with wind data. The basic principles of this technique are based on geostatistics. Altunkaynak (2005) then applied the TPCSV technique on wave data. A close investigation of this technique, however, shows that the technique was developed by giving more weight (degree of influence) to the distance between points of measurement than the changes in the regional variables with distance. It is believed that this can affect the performance of the approach. In addition, the TPCSV approach takes into consideration only analytical implications. In other words, the TPCSV technique is used to determine weighting coefficients analytically. This study was, therefore, undertaken to address the limitations indicated above by developing a technique that has the ability to make geometrical evaluations as well as analytical interpretations by giving more weight to the changes in the regional variables with distance as opposed to the TPCSV method. The technique proposed in this study is called Slope Point Cumulative Semi-Variogram (SPCSV).

2. Materials and methods 48

46041

2.1. Data used and analysis

WASHINGTON

46029

46005

46

46050 IDAHO

OREGON

44

46002 46027

PACIFIC

42

46022

OCEAN

40

46014

NEVADA

46013 46026

46059

46042

38 CALIFORNIA

36

46028

46062 46023 46011 46063 46064 46025

34

46047

-136

-134

-132

-130

-128

-126 -124 Longitude

-122

Fig. 1. Map of study area.

-120

-118

-116

Latitude

46006

In this study, the SPCSV method was developed to predict significant wave heights in January 2003 at 22 stations located in the Pacific Ocean off the west coast of the United States of America (USA) (Fig. 1). The stations are located at depths ranging from 47 m to 4599 m. In the study area, while larger significant wave heights appear in the winter season, relatively smaller waves occur in the summer season. The annual spatial average significant wave height at the study area is 2.31 m. Previously, Altunkaynak (2005) and Altunkaynak and Ozger (2005) applied the TPCSV and PCSV techniques do determine significant wave height using data from the same stations. The same data were used in the method developed in this study, i.e., SPCSV. In addition, the PCSV and TPCSV approaches were applied to determine the significant wave height using the same data with the objective of comparing the performance of these two techniques with the SPCSV method.

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A. Altunkaynak / Ocean Engineering 106 (2015) 220–226

For this purpose, the significant wave height predicted using the new approach was compared to the observed significant wave height and the significant wave heights predicted by the PCSV and TPCSV models. This comparison was performed by using Relative Error as evaluation criterion. 2.2. Description of the models 2.2.1. Point Cumulative Semi-Variogram (PCSV) approach The PCSV technique is a technique that makes spatial estimations by considering the spatial variability of the field. The concept of PCSV was first introduced by Habib (1993) and Sen and Habib (1998). The essence of the methodology relies on the weighting function obtained from joint evaluation of neighboring stations. The weighting function is determined based on distance (d) between a pivot station and other stations, and half of the square of the differences (γðdÞ) of the variables at the pivot station and other stations. In the PCSV technique, the value of a variable at a pivot station is estimated assuming that the values of the variables at the other stations have impacts on the variable at the pivot station. However, their impact is thought to decrease gradually with the increase in the distance between the pivot station and the other stations. The steps given below need to be followed to obtain a sample PCSV for pivot stations: (1) determine the Euclidean distances between the pivot station and all other stations, (2) calculate half of the square of the differences of variables as γðda Þ ¼ ð1=2ÞðJ p J a Þ2 , where Jp and Ja are regionalized variables (ReVs) at a pivot and neighboring stations, respectively, and d is the distance between these stations, (3) sort the γðdÞvalues with respect to distances in ascending order, (4) plot a graph as γðdÞversus the distances and (5) take cumulative sum of the nP 1 γðdÞvalues asγðdÞ ¼ γðdai Þ. Once the PCSV is developed, the regional dependency ifunction can be obtained by simply subtracting the standardized values of the PCSV from a unit value.

spatial prediction of streamflow. In addition, optimal regional dependency function was used to estimate streamflow in the study by Altunkaynak (2009). In this study, the spatial correlation between stations was determined as γðdÞ. However, in the calculation of values of γðdÞ, uniform distribution of nodal points should be achieved to maximize the accuracy of the results. The value of half of the square of the differences (γðdÞ) at two stations (Eq. (2)) can be taken as the deviation or the amount of error between the values of the variable at two stations. If these differences are small, this means that the similarity between these two stations is high. Similarly, stations are considered to be dependent to each other. Stations also exhibit homogeneity when this difference is small. Let's represent the significant wave height value at a pivot station by Hp and the significant wave height value at a station with a known distance, d, from the pivot station by Hp þ d., Half of the square of the difference between the pivot station and the other stations (γðdi Þ is, therefore, 1 ðH p  H i Þ2 2

γðdi Þ ¼

γ st ðdi Þ ¼

ð1Þ

As it can be seen in Eq. (1), the distance variable (⧍di) is present in the terms given as nominator and denominator. This shows that the distance variable is given higher degree of influence (more weight) than the regional variable in TPCSV approach.

γðdi Þ γ max ðdÞ

ð3Þ

In the same manner, the values of distance, di, can be standardized using the maximum value, dmax, as: di dmax

ð4Þ

where (di)st is the standardized distance. Fig. 2 shows that the standard PCSV increases with distance. For this reason, the value of the angle is taken to be between 01 and 901. The standard PCSV consists of several broken lines connecting successive stations. If two points are chosen from Fig. 2, the differences in distances between two successive stations on the horizontal axis and the difference in the corresponding values on the vertical axis represented by Δd and Δγ, respectively, and these can be determined as: Δdi ¼ di þ 1  di 

ð5Þ



Δγ i ¼ γ di þ 1  γ ðdi Þ

ð6Þ

1 0.9

Regional Dependency Function (PCSV)

0.8

46041

0.7

Standard PCSV

Δdi cos α ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðΔdi Þ þ ðΔγðdi ÞÞ

ð2Þ

here, Hi is the magnitude of a regional variable (ReV), which is the significant wave height in this case as mentioned above and di is the distance between the pivot station and the other station. The dependency function is a curve that shows the change in PCSV with respect to distance. Standardization of the dependency function is carried out to keep the numeric values of γðdi Þ between zero and one. This standardization (st) preserves the data characteristics while scaling the particular numeric values using the maximum value of γðdi Þ represented by γ max ðdÞ as:

ðdi Þst ¼ 2.2.2. Trigonometric PCSV (TPCSV) The TPCSV method is an enhanced version of the PCSV method in terms of obtaining the weighting coefficients. The TPCSV method, first proposed by Sahin and Şen (2004), considers the relationship between successive stationsto obtain the weighting coefficients. This relationship is explained and quantified as trigonometric function. TPCSV differs from PCSV in the determination of the weighting coefficients. However, the five steps mentioned above to obtain the PCSV diagram are repeated in the TPCSV method as well. The weighting coefficients in the TPCSV method are calculated by considering two successive points on PCSV diagram and the angle between them. The values of angles that are close to 01 show the presence of high dependency between the two successive points. On the other hand, higher angle values, for example 901, imply the independency between two successive points in the PCSV diagram of the TPCSV method. Weights are determined by simply taking the cosine of the angle α (cos αi) between two successive points, and this is given as:

ði ¼ 1; 2; 3…:nÞ

0.6 0.5 0.4 0.3 0.2 d

0.1 0

2.2.3. Slope Point Cumulative Semi-Variogram The Slope Point Cumulative Semi-Variogram (SPCSV) methodology was first introduced and used by Altunkaynak (2008b) for

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Standard distance

Fig. 2. Regional Dependency Function (PCSV).

0.8

0.9

1

A. Altunkaynak / Ocean Engineering 106 (2015) 220–226

Δγðdi Þ ðDf Þi ¼ tan αi ¼ Δdi

Δdi Δγðdi Þ

ð8Þ

and the sum of the cot α values is used to obtain the weighted average results as depicted in Eq. (9). T¼

n X

cot αi

ð9Þ

i¼1

where T represents sum of the weighting coefficients. The prediction function can then be defined as: Hp ¼

n 1X H cot αi Ti¼1 i

Regional Dependency Function (Slope PCSV)

35

46041

30 25 20 15 10 5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Standard distance

Fig. 3. Regional Dependency Function (SPCSV).

ð7Þ

If tan α is small, then the similarity between the two points will be high. This also means that the station significantly contributes to the pivot point. A large slope indicates that the similarity between the points and the contribution of the other point to the pivot point is very low. Since an inverse relationship exists between slope and weighting coefficient (Wc), the weighting coefficient can be taken as cot α, which is given as: cot αi ¼

40

tan ( )

The slopes of these broken lines give some indications about the similarity and dependency between the two points. The ratio of half of the square of the difference between the two successive points, ΔγðdÞ, to the horizontal distance between them gives the slope of the line (tanα). This value is also the first derivative of the function of the plot. The steeper the slope between the two points, the less similar the two stations are. In this study, the slopes of the broken lines which consist of the standard PCSV are interpreted. In doing so, either the change in γðdi Þ or the change in the distance is considered. Being close to each other does not necessarily imply that two stations have similar properties. However, the square of the value of the error between two closest stations ðH p  H d Þ2 should be low. It can be said that a station is similar and has a greater contribution to a pivot station when its distance from the pivot station and the relative change of the value of the variable are small. The SPCSV can be defined as dependency factor (Df), which is given as:

223

ð10Þ

where Hi is the magnitude of the regional variable (ReV, which is significant wave height in this study) and Hp is the predicted significant wave height. The advantages of the SPCSV approach over the other geostatistical techniques (Altunkaynak, 2008b; 2009) and its differences from these techniques can be given as below. 1. The standard PCSV function seems to be an increasing function as depicted in Fig. 2. Therefore, it is difficult to estimate the similarity of other points to the pivot point. RDF obtained by the SPCSV method shown in Fig. 3 is in a good agreement with the fluctuations. This means that the similarity of the other points to the pivot point increases as RDF becomes close to the x-axis and the similarity decreases when it goes far from the xaxis. 2. Even the smallest dependency of a station with the pivot point can be shown in RDF (Fig. 3) according to the SPCSV method. However, it is very difficult to show such dependencies by using the SV and PCSV approaches. 3. In the SV technique, the distances between coupled points are constant. However, the SPCSV technique can be safely applied on equally or non-equally spaced data sets. Therefore, this makes SPCSV to be more flexible. 4. Although PCSV is used for determining Regional Weighting Function (RWF), weighting coefficient of the closest point to the pivot point is “1” and the farthest one is “0”. This means

that the dependency function depends on the location rather than the magnitude of the regional variable. However, the most important feature of SV is that it depends on both the magnitude and location of a variable scattered in space. In other words, there is a limiting assumption which indicates that, in the PCSV method, the contribution of closer points to the pivot point for point estimation is the highest and the contribution of points located at far distances decreases. However, there is no such a restrictive assumption in the SPCSV method. 5. In the PCSV technique, natural characteristic of the data is corrupted due to cumulative calculation of the γ(di) values. However, in the SPCSV technique, some beneficial outcomes can be obtained from the calculation of the slope between two points by assessing the physical properties of the region or the points. 6. In the SPCSV method, each point can be evaluated and interpreted in detail. Unlike in the SPCSV method, in SV technique, the ReV can only be analyzed generally because of the absence of the concept of pivot point. 7. In the TPCSV method, the distance variable is given a higher degree of influence than the regional variable. As opposed to this, in the SPCSV method, the distance variables is found only in the denominator and only the variation of the regional variable with the distance between two stations is considered. Therefore, the SPCSV based prediction is unbiased.

The following steps are necessary for the successful application of the SPCSV approach. 1. As a first step, a pivot point (Hp) is defined or selected. 2. Then, coordinates are assigned to the pivot point and to the other stations using x–y pairing as (xp, yp) and (xi, yi), respectively. The distances between the defined pivot point and all qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the other stations are calculated as dpi ¼ ðxp  xi Þ2 þ ðyp yi Þ2 . 3. Half of the square of the difference of the values between the pivot point (Hp) and the other points are calculated to determine γ(d) using Eq. 2. 4. γ(di) values are sorted in ascending order according to the distance and cumulative γ(di) values. Δd and Δγ values are then standardized using Eqs. (3) and (4). 5. Standard PCSV function (plot) is developed by taking the standardized distances ((di)st) on the x-axis and the standardized cumulative values (γ(di)st) on the y-axis as shown in Fig. 2.

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A. Altunkaynak / Ocean Engineering 106 (2015) 220–226

40

Regional Weighting Function (Slope PCSV)

35

46041 30

cot ( )

25

20

15

10

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Standard distance Fig. 4. Regional Weighting Function (SPCSV).

6. The PCSV function is composed of many points with a line connecting each point to a neighboring point. Variations in the slope of the lines with the standardized distances are obtained by making use of RDF as shown in Fig. 3. 7. Variation in cot α value of the same line with the standardized distance constitutes of the RWF as shown in Fig. 4. 8. Predictions or estimations are then made by using Eq. (10). 9. The numbers of adjacent stations are, then, determined by considering the prediction error. This means that the prediction of the regional variable at the pivot station is undertaken based on at least two or more combinations of stations. Then, the number of combination of stations with the smallest prediction error value is taken as the number of adjacent stations. 3. Results and discussion A spatial function called SPCSV was used to predict the significant wave height for 22 stations located in the Pacific Ocean off the west coast of the USA as shown in Fig. 1. Selection of a pivot station is necessary prior to the application of the SPCSV method and, therefore, Station 46041 was selected as a pivot station in this study. Table 1 presents a summary of results based on the steps given above for the successful application of the SPCSV approach for Station 46041 (the pivot station). Fig. 2 shows that the standardized PCSV function of Station 46041 increases with distance. As mentioned earlier, this function consists of several broken lines. The slope ( tan α) of each line gives information about the similarity between the stations, and the inverse of the slope (cot α) can be used as a weighting coefficient that is used to calculate the contribution of a station to the pivot station. In Table 1, the 2nd column depicts the distances between the pivot station (Station 46041) and the other stations, and the 3rd column shows the standardized values of these distances. The 4th column presents the Dfs values, the 5th column contains Wcs and the 6th column depicts the contributions of the other points to the pivot station. In addition, Table 1 shows the predicted significant wave heights, relative errors and the number of adjacent stations. As can be seen from Fig. 2, the standardized PCSV function of Station 46041 has three regions. The transitions between these regions are very sharp. These sudden shifts reflect the change from one wave climatic region to another and, therefore, there are different wave climatic conditions in these regions. The standardized PCSV function (Fig. 2) increases rapidly in the first region between the standardized distance values of 0.1 and 0.45. The standardized PCSV function continues to increases at a high rate in the second region and at a reduced rate in the third region between

standardized distance values ranging from 0.45–0.90 to 0.90–1, respectively. The range of the standardized distance of the third region is narrower than those of the first and the second regions. The first and the second regions have wide ranges and cover nearly 90% of the total region. This wide region is evaluated in terms of wave energy potential. These evaluations are important for deciding the type of wave energy converter (devices that are used to generate energy from wave) that might be deployed in the region, and for optimizing management policies. Taking Fig. 3 into consideration, one can clearly see the presence of two parts. The first part lies between 0.1 and 0.90 standardized distances, and the second part lies between 0.90 and 1 standardized distances. In Fig. 3, values that lie closer to the x-axis indicate those stations that have similar features, while values farther away from the x-axis show those stations with dissimilar features. This identification can only be clearly and practically determined by using the function expressed in the SPCSV method. Fig. 4 shows the slope-based regional weighting function. When this function is closer to the x-axis, the weighting coefficient gets smaller and when the function is farther from the x-axis, the weighting coefficient becomes larger. The influence of radius, the amount of wave energy and the type of wave energy converters can be determined with the help of these curves. Optimum wave energy production and interpretation of results are very important in ocean engineering for planning, design and operation of structures. Let us assume that the data of Station 46041 is missing. First, the distances between Station 46041 and the other stations are calculated and put in ascending order. Then the distances are standardized by dividing every value by the maximum value (Eq. (4)). It can be seen from the 2nd column of Table 1 that the maximum distance is found between Stations 46041 and 46066. Dfs are calculated in column 4 of Table 1 using Eq. (7). The 5th column includes values of Wcs that was calculated using Eq. (8) and the contribution of each station to the pivot station is presented in the 6th column. The significant wave height value at Station 46041 is predicted from the three closest stations with the lowest relative error. The predicted significant wave height value at this station is H ¼3.20 m, the relative error is 0.42%, the number of adjacent stations is 3 and the influence radius is R¼ 302633.1 m. Table 2 presents the predicted values of the significant wave height using the PCSV, TPCSV and SPCSV methods, and the corresponding relative errors and the number of adjacent stations. In Table 2, the 2nd column includes observation values and columns 3–5 present the significant wave height values predicted by PCSV, TPCSV and SPCSV, respectively. Columns 6–8 present the corresponding relative errors of the methods indicated above. The numbers of adjacent stations that best predict the pivot station using these three methods are given in columns 9–11. The star symbol (*) in the 8th column indicates that the results predicted by the SPCSV method are more accurate than that of the PCSV method. This occurred at 19 stations out of the 22 stations. The mean relative errors of the SPCSV and the PCSV methods were found to be 6% and 10.4%, respectively. In addition, column 6 of Table 2 indicates that 8 out of the 22 relative error values of the PCSV method are greater than 10%. This number is found at 4 out of the 22 stations when using the SPCSV method. The ratio of improvement is, therefore, 50%. Moreover, SPCSV outperforms TPCSV when the same comparisons are made. SPCSV predicted the significant wave height better than TPCSV at 17 stations out of the 22 stations. The relative errors of the SPCSV and the TPCSV methods were found to be 6% and 8.7%, respectively. The numbers of stations that have relative error values greater than 10% are 4 and 7 using SPCSV and TPCSV, respectively. By taking relative error values into consideration, it can be stated that PCSV and TPCSV have close estimating performances. PCSV and

A. Altunkaynak / Ocean Engineering 106 (2015) 220–226

225

Table 1 Station 46041 detailed prediction calculations according to Slope PCSV. Station ID (1) Distance (m) (2) Standard distance (3) Regional dependency function Regional weighting function

46041 0 0 46029 137195.7 0.0828 46005 144861.6 0.0874 46050 302633.1 0.1827 46002 536666.7 0.3240 46066 596006.9 0.3598 46027 610455 0.3685 46006 729025.4 0.4401 46022 735656.9 0.4441 46014 904486.4 0.5460 46013 1018640 0.6149 46059 1040957 0.6284 46026 1075434 0.6492 46042 1191058 0.7190 46028 1310056 0.7908 46062 1394121 0.8416 46011 1420791 0.8577 46023 1436955 0.8674 46063 1492941 0.9012 46054 1495284 0.9027 46025 1509826 0.9114 46047 1656533 1.0000 Significant wave height prediction restricted sites Relative error (%) Number of adjacent sites Influence of radius

Dependency factors tan(α) (4)

Weighting coefficients cot(α) (5)

0 0.025146 16.20364 0.000206 0.224946 1.204783 4.196616 2.524405 0.611944 0.038678 0.496314 0.052821 2.129798 0.505968 0.314662 0.847825 0.001474 4.763609 1.072444 39.88173 29.09702 0.559129

0 39.7685 0.0617 4862.3402 4.4455 0.8300 0.2383 0.3961 1.6341 25.8547 2.0149 18.9320 0.4695 1.9764 3.1780 1.1795 678.2001 0.2099 0.9324 0.0251 0.0344 1.7885 3.20 m 0.42 3 302633.1 m

Significant wave height contributions (6)

0 133.6506 0.2551 15515.7209 16.9390 3.2456 0.6094 1.8409 4.9648 77.3870 5.1964 62.4082 1.1702 5.0772 8.5689 2.9520 2185.5138 0.5196 2.3876 0.0602 0.0513 4.3847

RDF: Regional Dependency Function. RWF: Regional Weighting Function. Dfs: Dependency Factors. Wcs: Weighting Coefficients. Table 2 Comparison of PCSV, TPCSV and Slope PCSV methods. Station ID (1)

46002 46005 46006 46011 46013 46014 46022 46023 46025 46026 46027 46028 46029 46041 46042 46047 46050 46054 46059 46062 46063 46066 Average

Wave height

January 2003 significant wave height prediction

Relative error (%)

Observation (2)

PCSV (3)

PCSV (6)

3.81 4.13 4.65 3.22 2.58 2.99 3.04 2.48 1.49 2.49 2.56 2.70 3.36 3.21 2.57 2.45 3.19 2.40 3.30 2.50 2.56 3.91

3.39 3.46 3.14 2.52 2.84 2.97 3.17 2.66 2.53 2.77 3.40 2.69 3.39 3.45 2.59 2.48 3.43 2.52 2.94 2.65 2.65 3.57

TPCSV (4) 3.34 3.61 3.24 2.54 2.77 2.94 3.01 2.55 2.55 2.71 3.00 2.61 3.39 3.30 2.59 2.48 3.37 2.44 2.84 2.49 2.56 3.62

Slope PCSV (5) 3.42 3.88 3.50 2.62 2.57 3.02 2.99 2.50 2.56 2.50 2.58 2.54 3.36 3.19 2.58 2.47 3.21 2.48 2.97 2.52 2.57 3.77

10.91 16.32 32.53 21.66 9.13 0.77 5.66 6.91 41.04 10.09 24.72 0.21 0.86 7.03 0.78 1.26 6.86 4.65 10.72 5.70 3.26 8.82 10.4

TPCSV (7) 12.37 12.76 30.38 21.03 6.76 1.75 0.96 2.96 41.36 8.15 14.68 3.32 0.86 2.89 0.69 1.03 5.41 1.61 13.76 0.68 0.12 7.33 8.7

Number of adjacent site Slope PCSV (8) a

10.37 6.05a 24.69a 18.68a 0.38a 0.98 1.53a 1.14a 41.66 0.48a 0.93a 5.79 0.03a 0.42a 0.28a 0.94a 0.50a 3.27a 9.91a 0.49a 0.31a 3.64a 6.0

PCSV (9)

TPCSV (10)

Slope PCSV (11)

3 8 5 5 4 3 7 6 4 6 6 4 5 10 3 5 5 2 7 6 4 3 5.0

7 8 5 9 9 3 3 4 4 3 9 3 3 3 3 9 9 7 7 9 6 7 5.9

7 6 3 5 4 3 3 2 4 6 6 5 8 3 3 2 4 2 5 3 6 5 4.3

PCSV: Point Cumulative Semi-Variogram. TPCSV: Trigonometric Point Cumulative Semi-Variogram. Slope PCSV: Slope Point Cumulative Semi-Variogram. a

Slope PCSV method has better predictions than PCSV.

TPCSV have relative errors of 10.4% and 8.7%, respectively. These results show that the SPCSV method outperforms the PCSV and TPCSV techniques. In the SPCSV method, the mean number of

adjacent stations required to provide data for estimating the missing variable at the pivot station is found to be 4, while the mean number of adjacent stations for the same purpose is found

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A. Altunkaynak / Ocean Engineering 106 (2015) 220–226

to be 5 and 6 for the PCSV and TPCSV methods, respectively. The performance of the SPCSV method in comparison with that of the PCSV and TPCSV methods is presented by using numerical and graphical methods. 4. Conclusion A new method called SPCSV was developed in this study for predicting significant wave height. Twenty two (22) stations located off the west coast of the USA were considered for this purpose. The Regional dependency functions and weighting functions were determined for each station using data obtained from previous studies. The radius of influence was computed for each station and the SPCSV method was compared with the PCSV and TPCSV methods. The new method (SPCSV) was found to outperform the PCSV and TPCSV methods at 19 and 17 stations out of the 22 stations, respectively, in estimating significant wave height. The number of adjacent stations required to make meaningful estimations with the SPCSV method was also found to be less than the amount required with the PCSV and TPCSV methods. For this reason, the application of the SPCSV method is considered to be more efficient when compared to the other two methods. Moreover, the mean relative error of the SPCSV method (6%) was found to be less than the relative errors of the remaining two methods all over the stations in the time period considered in this study. It can be concluded from this study that information generated from the application of such a method can be used during the design and identification of locations of wave energy converters. References Altunkaynak, A., 2013. Prediction of significant wave height using geno-multilayer perceptron. Ocean Eng. 58, 144–153. Altunkaynak, A., 2009. Streamflow estimation using optimal regional dependency function. Hydrol. Process 23, 3525–3533.

Altunkaynak, A., 2008a. Adaptive estimation of wave parameters by Geno–Kalman filtering. Ocean Eng. 35, 245–1251. Altunkaynak, A., 2008b. Estimation of streamflow by Slope Regional Dependency Function. Hydrol. Earth Syst. Sci. 12, 1121–1127. Altunkaynak, A., 2005. Significant wave height prediction by using a spatial model. Ocean Eng. 32 (8–9), 924–936. Altunkaynak, A., Ozger, M., 2005. Spatial significant wave height variation assessment and its estimation. J. Waterw. Port Coast. Ocean Eng. 131 (6), 277–282. Altunkaynak, A., Ozger, M., 2004. Temporal significant wave height estimation from wind speed by perceptron Kalman filtering. Ocean Eng. 31, 1245–1255. Bidlot, J.-R., Holt, M.W., 1999. Numerical wave modelling at operational weather centers. Coast. Eng. 37, 409–429. Goovaerts, P., 1999. Geostatistics in soil science: state-of-the-art and perspectives. Geoderma 89, 1–46. Habib, Z., 1993. Objective analysis with point cumulative semivariogram in meteorology. Istanbul Technical University, Istanbul Turkey (Unpublished MSc. thesis). Krige, D.G., 1951. A statistical approach to some basic mine evaluation problems on the witwateround. J. Chimic. Min. Soc. S.-Afr. 52, 119–139. Makarynskyy, O., 2005. Artificial neural networks for wave tracking, retrieval and prediction. Pac. Oceanogr. 3 (1), 21–30. Makarynskyy, O., Makarynska, D., 2007. Wave prediction and data supplementation with artificial neural networks. J. Coast. Res. 22, 146–155. Matheron, G., 1971. The Theory of Regionalized Variables and its Applications. Ecole de Mines Fontainbleau, France. Matheron, G., 1963. Principles of geostastistics. Econ. Geol. 58, 1246–1266. Paplinska-Swerpel, B., Paszke, L., 2006. Application of neural networks to the prediction of significant wave height at selected locations on the Baltic Sea. Arch. Hydro-Eng. Environ. Mech. 53 (3) 183–20. Roulston, M.S., Ellepola, J., Hardenberg, J.,V., Smith, L.A., 2005. Forecasting wave height probabilities with numerical weather prediction models. Ocean Eng. 32, 1841–1863. Saetra, O., Bidlot, J.-R., 2002. Assessment of the ECMWF ensemble prediction system for waves and marine winds. ECMWF Tech. Memo. 388. Sahin, A.D., Şen, Z., 2004. A new spatial prediction model and its application to wind records. Theor. Appl. Climatol. 79 (1–2), 45–54. Sen, Z., Habib, Z., 1998. Point cumulative semivariogram of areal precipitation in mountainous regions. J. Hydrol. 205, 81–91. Sen, Z., 1989. Cumulative semivariogram model of regionalized variable. Math. Geol. 21, 891–903. Soares, C. Guedes, Scotto, M., 2001. Modelling uncertainty in long-term predictions of significant wave height. Ocean Eng. 28 (3), 329–342. Tu, J.V., 1997. Advantages and disadvantages of using artificial neural networks versus logistic regression for predicting medical outcomes. J. Clin. Epidemiol. 49, 1225–1231.