Automated classification of the atmospheric circulation patterns that drive regional wave climates

Introduction Conclusions References


Introduction
Wave climates are strongly linked to atmospheric circulation.The link is complex and its direct functional description can be difficult to derive.However atmospheric circulation can be classified into discrete patterns (CPs), which represent different links to wave behavior.If these links can be clarified they can be used to asses changes in the wave Understanding the wave climate at a given region is of fundamental importance to coastal planners, managers and engineers.Coastal erosion depends strongly on the extreme wave events, which in turn are driven by atmospheric circulation.Therefore changes in circulation patterns can change wave climate parameters (wave height, direction etc).This has a direct impact on the location and severity of beach erosion.
Wave climates along coastlines dominated by a single storm system are easiest to define and changes in circulation patterns are reflected in wave observations.For example, Komar et al. (2010) were able to evaluate changes in the wave climate of the North Pacific and North Atlantic oceans.The shorelines along these coasts are typically dominated by one or two main storm systems with inherent seasonality.However in a diverse storm environment the trends can be more difficult to identify and quantify.
The aim of this study is to utilize existing statistical methods to identify the atmospheric circulation patterns driving the wave climate at a given location, with particular reference in extreme wave events.These events are defined here as periods during which significant wave heights exceed a threshold of 3.5 m.Physical links between atmospheric features and wave heights are complex and nonlinear.Therefore attempts to model wave characteristics derived from circulation patterns can be difficult and time consuming.Statistical knowledge gained from the observations of wave climates and pressure fields allow insight into this complex relationship without the need for explicit physical coupling.This can be a useful tool for risk analysis since it provides insight into the source of extreme events.If we understand the circulation patterns that drive extreme events, then their occurrence (or the occurrence of similar patterns) can have a degree of risk attached to it.For example the risk could be the likelihood of an extreme wave event, of severe erosion, of extended storm durations, or a combination of all three.
Circulation patterns are herein described in terms of pressure anomalies on the 700 hPa geopotential.The types or classes of anomalies can be specified by two approaches: (1) those specified prior to classification; (2) those that are derived and evolve during the classification process (Huth et al., 2008).In the past, anomaly pat-Figures

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Full  (Lamb, 1972;Hess and Brezowsky, 1952;Huth et al., 2008).However the power of modern computers provides a means to generate numerical solutions to complicated algorithms that can automate the process.It is important to note that atmospheric circulation patterns are not a set of separated, well-defined states.CPs change smoothly between states that form part of a continuous sequence of events (Huth et al., 2008).Therefore the classes (or types) merely represent simplified climatic events responsible for specific variables of interest.While automated derivations of classification types utilize objective reasoning, according to Huth et al. (2008) the procedure as a whole cannot be considered fully objective.A number of subjective decisions are still employed.For example the number of CPs to use and the method of differentiating classes.
The classification method used for this study is a fuzzy rule based algorithm developed by Bárdossy et al. (1995).The algorithm was originally used to link atmospheric CPs with rainfall events (Bárdossy et al., 2002).In the present study the method has been adapted to use wave heights to guide the classification of CPs.The main aim of this paper is to investigate the feasibility of using this method to identify CPs that are the main drivers of regional wave climates for application to coastal vulnerability assessments.

Case study site
The KwaZulu-Natal (KZN) coastline (Fig. 1) has a high energy wave climate.Tropical cyclones, mid-lattitude (extra-tropical) cyclones and cut-off lows have been cited as important drivers of the local wave climate (Mather and Stretch, 2012;Corbella and Stretch, 2012b;Rossouw et al., 1982).Tropical cyclones that become stationary to the south east of Madagascar can occasionally drive large wave events that cause severe Figures

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Full beach erosion in KZN (Mather and Stretch, 2012;Corbella and Stretch, 2012b).Cutoff lows are deep low pressure systems that are displaced from the normal path of west-east moving mid-latitude cyclones.Instabilities within the westerly zonal flow, due to the high wind shear, create vortices (low and high pressure), which can become cut-off and move equatorward (Preston-Whyte and Tyson, 1988).This diverse storm environment leads to seasonality within the wave climate.On average autumn and winter are associated with the largest wave energy, while summer has the smallest (Corbella and Stretch, 2012b).Seasons are defined according to Table 1.

Classification methods
The aim is to identify a classification in which a set of classes defining atmospheric pressure fields can explain the occurrence of wave events at a specified location.There are many ways in which classification algorithms can be constructed.Classifications can be subjective, objective or a mixture of both (Bárdossy, 2010;Huth et al., 2008).The set of CP classes can be derived and linked to local weather phenomena through an optimization procedure (see Sect. 2.3 and Bárdossy, 2010).Classes linked with these variables explain, as best possible, their occurrences.This is a useful tool in guiding the algorithm to an optimal solution.Classifying CPs linked with extreme wave events is the focus of this study.The classifying procedure uses wave heights as the dependent variable to find classes of the independent variable, atmospheric pressure anomalies.The method is described in detail by Bárdossy et al. (1995);Bárdossy (2010).Only a brief overview is given here.
The classification is based on daily normalized anomalies, derived from the 700 hpa geopotential height with a grid resolution of 2.5 • (10 t, then the anomaly at location i and time t is given as where k(i ) and σ(i ) are the average and standard deviation of the geopotential at location i .Each CP is defined by a rule which comprises of a number of fuzzy set membership functions.The n th CP is described by the fuzzy rule n as a vector for all available grid points (1, . . ., K ), where V is the matrix containing all CP rules (1, . . ., n), the index V (i , n) is the fuzzy set number corresponding to the location i for rule n.The rules consist of the following fuzzy sets: 1. Fuzzy set number 0 -any type of anomaly, 2. Fuzzy set number 1 -strong positive anomaly, 3. Fuzzy set number 2 -weak positive anomaly, 4. Fuzzy set number 3 -weak negative anomaly and 5. Fuzzy set number 4 -strong negative anomaly.
The fuzzy set numbers (1, . . ., 4) describe the locations of different pressure types, however the fuzzy set number 0 is irrelevant for the CP classification.The algorithm only considers patterns with structures corresponding to the arrangement of the fuzzy set numbers 1, . . ., 4.
From the fuzzy set numbers described above, a membership grade µ at location i can be assigned for each daily anomaly pattern as where n is the fuzzy rule, g(c n (i ), t) is the membership function for the fuzzy set number j at location i at time t (Bárdossy et al., 2002).It follows that a combination of the Introduction

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Full membership grades provide insight into the performance of each CP rule in relation to the daily anomaly patterns.A degree of fit (DOF) is computed for each CP rule and the rule with the highest DOF value is assigned to the circulation pattern for that day.The degree of fit is defined as follows (Bárdossy et al., 1995(Bárdossy et al., , 2002)): where c n is the CP rule n, t is the day, µ n (i , t) is the membership grade at location i for rule n, N(n, j ) is the number of grid points corresponding to the fuzzy set number j for fuzzy rule n, and the exponent P j is a parameter that allows us to emphasize the influence of selected rules on the DOF.
The CP rules were obtained via an optimization procedure following Bárdossy et al. (2002) and is described in Sect.2.3.

Optimization methods
The goal of the optimization is to derive a set of CP rules defining dominant circulation patterns in a particular region.The rules are strongly linked to a variable of interest.
The optimization procedure should maximize dissimilarity between the CP types while minimizing the variability within the classes.The significant wave height (H s ) is the "guiding" variable for this study.The algorithm considers both the daily average significant wave height and the daily maximum significant wave height.Wave data was obtained from the various wave buoys located along the KwaZulu-Natal coastline for the period 1992-2009.A comparison of the data sets by Corbella and Stretch (2012b) showed a strong correlation between the wave buoys.The optimization procedure was carried out for the period containing all wave data (1992-2009) A simulated annealing algorithm following Aarts and Korst (1989) is used in the optimization procedure.Details of the optimization procedure is given in Bárdossy et al. (2002).A general de-Introduction

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Full scription will be given here.The optimization algorithm is briefly described following Bárdossy et al. (2002) as: (0) Randomly assigned CP rules are initialized and their performance is evaluated through an objective function O.
(1) The initial "annealing temperature" is set to q 0 .
(2) A rule n is selected randomly.
(3) A location i is selected randomly.
(4) A fuzzy number c * is selected randomly.
(12) Decrease the "annealing temperature" such that q new < q old .
(13) Repeat the steps 2 -12, until the number of accepted changes becomes less than a predefined limit.
The optimization relies strongly upon a set of objective functions.The objective functions are based on the extreme wave events, wave heights and storm duration as discussed in Sect.2.3.1.Introduction

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Objective Functions
A good classification contains classes with corresponding wave statistics which differ from the statistics calculated without classification.The goal of the classification is to obtain a set of rules which correspond to the occurrence of extreme waves.Therefore the objective functions used within the algorithm are designed to optimize the CP occurrences which coincide with extreme wave events.These are relatively rare events.A random classification leads to the same probability of occurrence as the mean for each rule, which is undesirable (Bárdossy, 2010).A good classification should lead to rules that differ from the climatological mean for the selected variable, in this case the wave height.
The first objective function is applied at the time of the maximum significant wave height during the extreme wave events (i.e. when H s ≥ 3.5 m): where θ is a predefined threshold (θ = 3.5 m in this case), S is the number of stations, T is the total number of days, p(CP(t)) i is the frequency that the threshold is exceeded at station i for a given CP on a day, p i is the unclassified probability of exceedance for all days in period T .
The second objective function incorporating average significant wave heights is defined as where z(CP(t)) i is the mean significant wave height on a day with a given CP at station i and z i is the mean daily wave height without classification at the same station i .Storm Figures

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Full durations are defined as the times from when wave heights exceed 3.5 m to the time when they again reduce below 3.5 m.These storms are represented in Eq. ( 4) since they occur simultaneously with the threshold exceedances.
A linear combination of Eq. ( 4) and Eq. ( 5) was used to optimize the solution to the classification algorithm.

Classification quality
Classification quality refers to the ability of the algorithm to maximize dissimilarity between a set of CP classes, while minimizing variability within daily CP realizations.This study focuses on classifying CPs driving extreme wave events, therefore there are two criteria for measuring the classification quality.The first is the ability of the classification to explain extreme wave events.The second is the variability of the classifications within each CP class.There exists an optimal number of CP rules which successfully explain extreme events and daily CP realizations.Too few rules implies that the resulting CPs do not allow a proper distinction of the causal mechanisms and would lead to classes which have statistics similar to the unclassified case.Too many classes increases the computational effort and captures features that are not general and do not correspond to the wave generating mechanisms.Bárdossy (2010) suggests utilizing the objective functions as a measure of the classification quality.Huth et al. (2008) list a number of different quality measures that explain the separability between and variability within CP classes.For this study the variability within the classes as well as the degree of fit are used as measures of the classification quality.This provides insight into the performance of the classes with respect to their ability to explain average CPs.
The variability of extreme events is defined as the position of the lowest anomaly relative to the average pattern.This was assumed as the storm centre.Wave events are driven by storms associated with low pressures (i.e.negative anomalies).The performance of the CP classes in explaining extreme wave events can be measured by their relative contribution to extreme events, namely the ratio p n /p where, p n is the probability that the threshold is exceeded for a given CP rule n and p is the unconditional 1136 Introduction

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Full probability that the threshold is exceeded for all events A classification strongly linked to the wave climate should define classes whose frequency of occurrence correspond to the average and extreme wave events.This implies that CPs driving extreme events should occur infrequently, whereas CPs driving the average wave climate should occur more frequently.

Dominant CP classes
The objective functions (Eqs. 4 & 5) were used to derive a set of CP classes which explain extreme wave events.Figure 2 shows the average anomaly patterns for all the CP classes.CP99 refers to an unclassified class.Useful statistical parameters relative to this study for a given CP class are: (a) frequency of occurence, (b) percentage contribution of extreme events, (c) average and maximum significant wave heights (H s ).These parameters are obtained from the classification and are shown in Table 2.
The results show two trends in CPs that drive wave development.Firstly CP01 and CP02 (Fig. 2a and b) according to Table 2 occur most frequently (∼ 17 % of the time).
CP01 resembles that of mid-lattitude cyclones which frequently travel in a west to east direction south of the country, while CP02 resembles the high pressure systems that follow the mid-lattitude cyclones.Secondly, Table 2 shows that CP03 is associated with 30-60 % of all extreme wave events.The large contribution by this class to extreme events is present all year round with the highest contribution in winter (∼ 65 %).CP03 (Fig. 2c) occurs infrequently (7-9 % of the time), but when it does occur it is associated with average and maximum significant wave heights ranging from 2.4-3.0 m and 5.0-8.5 m respectively.CP05 and CP06 (Fig. 2e and f) according to the classification are responsible for about 30 % of extreme events in spring and summer respectively.CP06 represents low pressure anomalies southeast of Madagascar.This appears to resemble the strong low pressure systems that are associated with tropical cylones.Introduction

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Full According to Mather and Stretch (2012) low pressure systems southeast of Madagascar can cause large swells.CP05 resembles low pressure systems over the interior and which extend southwards.
No time lag was considered when deriving the CP classes.This constrains the algorithm to only consider CPs occurring on the day of a wave event and assumes that extreme events are driven by relatively stationary CPs.

Degree of fit (DOF)
The degree of fit relates to how well the CP for each day is classified as a given class relative to the rule file.Figure 4 shows the average anomaly pattern for CP03 together with the CPs associated with both the highest and lowest degree of fit value for that class.CP03 is associated with cut-off lows to the east/south-east of South Africa.The pattern also shows a strong high pressure region to the southwest.The combination of strong cut off lows occurring in conjunction with high pressure regions are important features for channeling waves towards the eastern coastline.Figure 4c is the CP with the lowest degree of fit for the class CP03 and it shows only a weak anomaly pattern.

Variability within classes
It is expected that in the vicinity of the regions defining rule types (high or low pressures) the standard deviation should be low.This is because the classification is based on the location of these rules in comparison to the anomaly patterns for specific days.
Whereas the locations of "any anomaly" rule types (fuzzy number 0) are expected to have significant variability, the variability in the vicinity of negative anomalies (low pressures) can be attributed to the movement of the low pressure systems.It is also expected that high pressure systems are more stable with lower variability in their positions.For example Fig. 4d shows lower variability in the vicinity of the high pressure re-

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Full gion while high variability (standard deviation of 1) in the low pressure region.This can be attributed to the movement of low pressure systems around the negative anomaly.
Figure 4d shows high standard deviation values in comparison to the mean negative anomaly.This could also indicate that CP anomalies driving extreme events (cut-off lows) are associated with strong negative anomalies.

CP rules and extreme events
Daily CP's classified as in a certain class for extreme wave events (H s ≥ 3.5 m) were compared to the average patterns for that class.Figure 3 shows the average pattern for CP03 together with selected extreme events corresponding to CP03.The centres of the CP's are shown as "+" symbols in the plot.A centre is defined as the location of the peak negative anomaly.The variability within the class is apparent.However the majority of CPs classified as CP03 resemble strong cut-off lows to the east/southeast of the country.It is apparent that these strong cut-off lows drive extreme wave events.Figure 5 and Table 3 show CP's associated with the six largest significant wave height events.Four out the six events have been classified as CP03, the class contributing to the majority of extreme events.The concept of CPs belonging, to some degree, to all the classes is evident in Fig. 5f.This shows a similar pattern to CP04 and CP08, which both represent low pressures southeast of Madagascar.However according to the classification this CP belongs to class CP08 and not CP04.From visual inspection it appears to resemble class CP04 better than CP08. Figure 5a and c

Discussion
Classifying circulation patterns is a useful tool for investigating the occurrence of certain patterns over a given region.There are many different techniques used for classifying CPs, each of which has its benefits and drawbacks (Huth et al., 2008).Classification can be subjective or objective (to a degree).However the goal is always to group similar patterns into individual classes.A useful application for engineering purposes is utilizing a variable of interest to "guide" the algorithm to find CPs linked to its occurrence.Bárdossy et al. (2002) successfully implemented this to classify CPs that explained wet and dry events in Europe.
The emphasis of the present study has been on the statistical link between atmospheric circulation patterns and extreme wave events.This is the first time the method described here has been used in this context and it has the potential to improve current methods of risk analysis.The benefit of fuzzy logic as a classification tool is that each daily CP belongs, to some degree, to all the CP classes.This is characteristic of atmospheric circulation where daily CPs form part of a continuum rather than a set of individual states as suggested by the derived CP classes (Huth et al., 2008).However a potential drawback of the method is the manner in which the CPs on each day are assigned to a class (Huth et al., 2008).The degree of fit (Sect.3) used in this study incorporates the connectivity to a given class through AND/OR combinations of HIGH/LOW and NOT HIGH/NOT LOW anomalies as described in Bárdossy et al. (1995).This technique has been successful in associating CPs with rainfall events (e.g.Bárdossy et al., 2002;Bárdossy, 2010) In the context of our case study site on the east coast of South Africa the most frequent CPs are low and high pressure anomalies located south of the country.This can be attributed to the west-east progression of mid-latitude cyclones which frequent this area.They are major contributors to the wave climate along the South African coastline (Rossouw et al., 1982).The low pressure systems can become isolated after being displaced towards the equator and can become stationary (Preston-Whyte Introduction

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Full and Tyson, 1988).These stationary cut-off lows can drive the development of extreme wave events.Table 2 indicates that the dominant CP that drives extreme events along the KwaZulu-Natal coastline is CP03, which is associated with abnormally low pressure to the east-southeast.CPs classified as CP03 resemble cut-off lows and the pattern agrees with speculations by Mather and Stretch (2012); Rossouw et al. (1982); Corbella and Stretch (2012b) concerning drivers of extreme waves.Low pressure anomalies linked to storms east to south-east of South Africa drive wind fields that direct the wave attack toward the coastline.Callaghan et al. (2008); Corbella and Stretch (2013) highlight the importance of identifying independent storms for risk analysis of extreme wave events.One limitation with the methods described herein is that it is difficult to evaluate the independence of the different CPs.A particular storm in various stages of development may belong to a number of CP classes rather than a single class according to the classification scheme.Examples are cut-off lows that become detached from extra-tropical cyclones traveling west to east in the region south of the country.The process of storm development drives wave development.This suggests that it may be better to locate a specific type of CP at any location rather than a specified type of CP at a fixed location.

Conclusions
A fuzzy rule based classification method has been adapted to identifying the atmospheric circulation patterns that drive regional wave climates.The east coast of South Africa was used as a case study.The method is based on normalized anomalies in daily 700 hPa geo-potential heights.The CP classes are derived from an optimization procedure which is guided by a variable of interest, in this case wave heights.The classification shows a strong anomaly pattern east-southeast of South Africa which explains 30-60 % of extreme wave events.This CP type explains extreme events in all seasons.However it occurs infrequently (∼ 8 % of the time) and is associated with large wave heights ranging from 5.0-8.5 m.Frequently occurring CP classes have sim-Introduction

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Full ilar structure to mid-latitude cyclones or translational low pressure systems (followed by a zone of high pressure) that occur south of South Africa (Taljaard, 1967).This methodology discussed here appears to be new in the context of wave climate analysis and has potential for application to risk assessment studies in coastal management and engineering.Figures  Full  Full  Full Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | terns were identified by experts in the field: examples are the Hess-Brezowski catalogue or the Lamb classification Discussion Paper | Discussion Paper | Discussion Paper | heights were obtained from the ERA-Interim data set for the period 1979-2009 (http: //apps.ecmwf.int/datasets/).Let k(i , t) be the geopotential height at location i and time Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | and run the classification.(7) Calculate the new performance O * for the new rules.(8) If O * > O, accept the change.(9) If O * ≤ O, accept the change with probability exp O−O * q j .(10) If the change has been accepted, replace O by O * .
Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | are the CPs associated with the March 2007 storm which caused severe coastal erosion along the KwaZulu-Natal coastline (Mather and Stretch, 2012; Corbella and Stretch, 2012a) with significant wave heights reaching 8Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Fig. 1 .
Fig. 1.Locations of the wave observation buoys at Durban and Richards Bay, along the KwaZulu Natal coastline.

Fig. 1 .Fig. 2 .
Fig. 1.Locations of the wave observation buoys at Durban and Richards Bay, along the KwaZulu Natal coastline. 18

Fig. 2 .Fig. 3 .
Fig. 2. Average anomaly patterns for all CP classes: 1-8.Positive anomaly contours are shown as the dashed line while negative contours are solid. 19

Fig. 3 .Fig. 4 .
Fig. 3. (a) Average CP03 with (+) symbols indicating the centers of all negative anomalies (low pressures) contributing to the class.(b) and (c) show actual CP's for the dates 19 March 2007 and 30 August 2006 respectively, both of which were classified as members of the CP03 class. 20

Fig. 4 .
Fig. 4. Average anomaly pattern for CP03 (a) with (b) the anomaly with highest DOF, (c) the anomaly with lowest DOF value, while (d) shows the standard deviation for all CP03 anomalies.

Table 1 .
The allocation of months to seasons.

Table 2 .
CP Occurrence frequencies and wave height statistics associated with each CP class.
a CP99 is the unclassified class.Blank entries imply zero occurrences in the data set.Introduction

Table 3 .
Six of the highest extreme wave events on record and their associated CPs for the period 1992-2009.