Space–time series modelling of beach and shoreline data

Environmental Modelling & Software 16 (2001) 299–307 www.elsevier.com/locate/envsoft

Space–time series modelling of beach and shoreline data P.D. LaValle, V.C. Lakhan *, A.S. Trenhaile Department of Earth Sciences, School of Physical Sciences, University of Windsor, Windsor, Ontario, Canada N9B 3P4 Received 12 July 1999; received in revised form 3 November 1999; accepted 10 August 2000

Abstract This paper focuses on utilizing Box–Jenkins modelling procedures to identify models which best describe a time series (1978– 1994) of beach and shoreline data collected from the Northeast Beach, Point Pelee, Lake Erie, Canada. The results highlight the influence of localized stochastic processes on beach sediment flux levels and on shoreline change. A spatial autoregressive model best describes the variation for beach net sediment flux. The model reinforces the hypothesis that beach net sediment flux in adjacent sites are interdependent and represent the cumulative spatial effects of short-term sediment flow processes such as longshore drift and persistent localized sediment circulation cells. Additional findings also suggest that a space–time autoregressive model best fits the data on spatial–temporal variations of shoreline retreat. The implication of this model is that the shoreline will retreat quite rapidly, but readjusts to an aggradational state at a much slower rate.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Box–Jenkins; Beach variations; Canada; Spatial–temporal; Spatial autoregressive model

Software availability Program title: PROFILE PLOT Program language: FORTRAN ’77 Developer: Dr. Placido LaValle Contact address: Department of Earth Sciences, University of Windsor, Windsor, ON Canada N9B 3P4 First available: 1991 Hardware: PC 486 or higher with math co-processor Cost: $25.00 Canadian Program title: STATISTICA Availability: StatSoft Inc., 2300 14th Street, Tulsa, OK 74104

1. Introduction While coastal scientists and allied researchers are utilizing various statistical techniques to gain insights on the long-term temporal and spatial dynamics of the beach

* Corresponding author. Tel.: +1-519-253-3000 (ext. 2183); fax: +1-519-253-6214. E-mail address: [email protected] (V.C. Lakhan).

and nearshore systems there is, nevertheless, a paucity of research work on space–time series modelling of beach and shoreline data. From the literature it is known that several researchers have used the empirical eigenfunction technique to analyze large data sets that describe variations in beach profile configurations (Winant et al., 1975; Aubrey, 1979; Bowman, 1981; Birkemeier, 1984; Wood and Weishar, 1984; Fisher et al., 1984; Aubrey and Ross, 1985; Lins, 1985; Ostrowski et al., 1990; Pruszak, 1993; Hsu et al., 1994). While eigenfunction analysis can provide valuable insights on the dissimilarities of profile lines and their responses, the technique relies on interpretations regarding the processes influencing morphological changes. In addition, Hsu et al. (1994), p. 204) pointed out that “the physical interpretation relating to each eigenfunction for the coastal process depends on the data length and the characteristics of the study area.” In a study on the temporal analysis of shoreline recession and accretion Dolan et al. (1991) assessed the endpoint rate, average of rates, linear regression, and jackknife methods which are commonly used to analyze shoreline change. Although the authors preferred the rates-of-change methods because they are more likely to represent the best approximation of the true rates-ofchange they, nevertheless, stated that “deciding on which method to use in a given area should be based on the

1364-8152/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 4 - 8 1 5 2 ( 0 0 ) 0 0 0 8 6 - 4

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purpose of the investigation coupled with an understanding of the temporal variability of the system in question” (p. 741). Another statistical method, the theory of regional variables or geostatistics, has been used by Dolan et al. (1992). Geostatistics, based on the spatial continuity of natural phenomena, has been found to be effective to estimate the along shore rates-of-change and spatial variations along profile transects. Shoreline rates-of-change, which exhibit a nearest neighbor effect or spatial autocorrelation, can be highlighted with the theory of regionalized variables which uses the random function concept to model the influence of nearby points. “The autocorrelation phenomenon reflects the spatial continuity of the driving processes which produce a measurable response such as shoreline change” (Dolan et al., 1992, p. 264). A new method for predicting shoreline change from historical data was presented by Fenster et al. (1993). The authors proposed a procedure which uses regression techniques to determine the linear or nonlinear polynomial which best fits the historical data according to the Minimum Description Length modelling criterion of Rissanen (1989). While the technique will permit identification of significant changes in shoreline trends the authors stress that judgment of past and future shoreline movement is hindered by the quantity and quality of the available data. Spectral analysis techniques are also applicable for analyzing a time series of data from topographic profiles, and other natural time series (for example, river discharge, daily temperature, lake turbidity, river plan form) with a uniform sampling interval (Hegge and Masselink, 1996). The results of the spectral analysis represent the amount of variance of the time series as a function of frequency. The time series analysis techniques of parametric spectra, autocorrelation, and cross-correlation were utilized by Walton (1999) to investigate shoreline plan form advance and recession rates for a coastal segment of Florida. The analysis of the spatial data allowed for the identification of low frequency rhythmic patterns in the shoreline. The low frequency shoreline oscillations were found to be non-stationary in time. While different statistical techniques can be used to analyze beach and shoreline data it is, nevertheless, evident from the aforementioned literature review that few authors have concentrated on joint spatial–temporal modelling of beach and shoreline data. This paper will, therefore, focus on utilizing Box–Jenkins modelling procedures to identify models which best describe a long time series (1978–1994) of beach and shoreline data collected from the Northeast Beach, Point Pelee, Lake Erie, Canada. Before presenting the procedures and results of the Box–Jenkins modelling approach the paper provides a brief description of the study area and the data collection methodology.

2. The Northeast Beach study area Point Pelee is located south of the forty-second parallel, and extends 10 km into the western basin of Lake Erie (Fig. 1). Point Pelee contains a large marsh, and is flanked by beach systems, including the Northeast Beach. The Northeast Beach is characterized by periodic episodes of erosion. Consequently, a shoreline monitoring program was initiated in 1978 when an artificial berm-beach complex was constructed over a major breach in the barrier beach system (LaValle and Lakhan, 1997a). Coakley and Cho (1972) found that net shoreline erosion along the eastern beaches of Point Pelee had occurred since 1947. The erosion accelerated along the Northeast Beach in 1973 when a large breach developed 400 m south of the Point Pelee National Park boundary. As a result, a system of concrete tetrapods was constructed along the northern 300 m of the beach, and an artificial berm was constructed to fill the breach. In 1979, a sediment renourishment program was instituted along the central portion of the Northeast Beach. Also, in 1984, a large armor stone breakwater was constructed along Marentette Beach immediately north of the Northeast Beach. In spite of emplacing a 260 m long treble line of concrete tetrapods in the northern sector of the Northeast Beach to reduce shoreline erosion a massive embayment

Fig. 1.

The Northeast Beach at Point Pelee, Lake Erie, Canada.

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developed in 1985. To assess changes in the shoreline and morphology of the beach Kuznik et al. (1986) examined the collected data and indicated that the sediment changes along the profiles of the Northeast Beach exhibited distinctive patterns of erosion and degradation. Since the report by Kuznik et al. (1986), research by LaValle and Lakhan (1997b) has shown that sediment renourishment programs, construction and replacement of infrastructure erosional control devices, and high lake levels have combined to affect the spatial and temporal variability at the Northeast Beach. The beach and shoreline data (1978–1994) collected from the Northeast Beach are examined in this paper.

3. Data collection methodology

Semi-annual topographic and bathymetric data were collected by LaValle (1978–1994) over a predetermined grid established at the Northeast Beach by Setterington (1978). The data provide the measurements necessary to calculate beach net sediment flux (BSF) and net shoreline positional change (NSC). Beach net sediment flux is defined as the volume of net sediment change occurring over a six-month period for a one metre wide area extending from the back-beach limit to the lowest waterline observed in the six-month period. Net shoreline positional change is defined as the difference in shoreline position along the survey transect measured in the sixmonth period. The data for this paper were obtained biannually from eight profile transects (hereinafter referred to as profiles) on the Northeast Beach for all years between May 1978 and November 1994. The eight profiles (57A, 73, 146, 220, 293, 80N, 160 and 59A) shown in Fig. 1 are oriented perpendicular to a baseline, and are between Park Survey Monument 59A (north) and Park Survey Monument 57A (south). Profile 293 is located 261 m south of Park Survey Monument 59A, and 293 m north of Profile 57A. Profile 293 is also located 23.32 m from Survey Monument 58A at an azimuth of 96 degrees, but this survey monument was destroyed in 1991. However, the continuous addition of secondary markers in the backbeach area helped to maintain the integrity of the baseline. Data from the eight beach profiles, which are 74–80 m apart, were collected in May and November of each year. Prior to each survey, the original survey baseline is identified and resurveyed using a Sokkisha B2 Automatic level. Its operation was aided by the fact that the survey monuments lie on a straight line with an azimuth of 18 degrees. This baseline was established by Setterington (May, 1978) and reestablished by Murray

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(November, 1978). The location of key profile transect origins such as that for Profile 73, 293, 80N, and 160 were either delimited by wooden stakes or by other markers such as trees in the back-beach area, and these secondary landmarks were relocated when necessary using either a Lufkin survey chain and Brunton Compass or a Sokkisha B2 Automatic level. With the use of a Sokkisha B2 Automatic level, measurements along each profile were taken at five-metre intervals. These profiles were oriented perpendicular to the baseline defined between survey monuments 57A and 59A which were part of the survey monument system established in 1977 (Brewer, 1976).The beach was surveyed from the edge of the marsh to the water sediment interface. All measurements were taken under calm conditions or with very light offshore breezes. For the sake of accuracy and safety, measurements were taken when only ripples or waves less than 0.15 m were present. Although seiche effects in this lake average 0.03 m (Carter, 1988), care was taken to obtain measurements when these effects were not present during the surveys. Lake Erie has a very small tidal range averaging about 0.02 m with spring tides approaching 0.05 m, so tidal variations played a nearly insignificant role in the analysis. By extending the profile lines, data were also collected for at least 100 m lakeward. This was done using a motorized boat, a stadia rod to measure depths, and a Foresters Chainman II to measure distances from the shoreline. Care was taken to align the boat with two markers on the beach to keep the nearshore transect as nearly perpendicular to the baseline as possible. It should be noted the data derived from the nearshore zone contributed very little in the calculation of beach net sediment flux (BSF) and net shoreline positional change (NSC) in this study. Once the profiles were surveyed, they were plotted using a microcomputer plotting program titled PROFILE PLOT. This program, developed by LaValle (1991), is designed to plot two successive surveys for each profile, and to calculate the net gains or losses of sedimentary material along the profile line. By using elevation changes and surface area the mass of material gained or lost between observation times is calculated. The results are then expressed as beach net sediment flux (BSF) for each profile. For this study BSF represents the net topographic volume change that has taken place between the most lakeward shoreline position observed during the survey time interval, and the landward terminus of the beach for a one metre wide transect. In addition to BSF data, changes in shoreline position are obtained for each profile by determining the net difference in the distance of the shoreline from the baseline for successive surveys. Net shoreline positional change (NSC) can thus be taken as being the net displacement of the shoreline observed over the survey interval.

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4. Modelling procedures: the Box–Jenkins modelling approach Box and Jenkins (1970, 1976) proposed an entire class of models called autoregressive, integrated, movingaverage (ARIMA) models, that seems applicable to a wide variety of situations. In brief, standard Box–Jenkins univariate time series modelling is based on the serial autocorrelation between the observed values taken for a specified set of lag intervals (usually less than the number of observations divided by four). Usually, the autocorrelation coefficients are plotted against the lag intervals on a graph called a correlogram, and the resulting function is called the autocorrelation function (ACF). In addition, the partial autocorrelation coefficients are calculated for the same number of lag intervals where a partial autocorrelation for lag i represents the autocorrelation at lag i with the effects of the autocorrelations at lag intervals 1 through i⫺1 filtered out. (This is similar in nature to a partial correlation coefficient). The partial autocorrelation coefficients are plotted against the lag intervals on a partial correlogram depicting the partial autocorrelation function (PACF). Based on an examination of the ACF and the PACF a model is fit to the data. If the ACF decays exponentially to near zero, and the PACF shows a distinct spike at lag 1, then a long memory Markov process may characterize the data, and a first order autoregressive model AR(1) might adequately describe the data. Such a model would have the form: Zt=fZt⫺1±e where f is the autoregressive parameter and e is the residual or error term. If such a model is to be accepted then the e terms should not be autocorrelated. In some cases ‘transients’ or random shocks associated with short memory moving average processes (MA) may produce an ACF with a single significant spike at lag 1, and no significant autocorrelations elsewhere. In this case one may fit a first order moving average model which has the form: Zt=qet⫺1±et, where q is the estimated moving average parameter and et⫺1 is the residual at lag t⫺1 while et is the residual term. As mentioned previously, after the model is fitted, the residuals et should not be autocorrelated if the model is to be accepted. Complex correlograms and partial correlograms may suggest that a combination of autoregressive (AR) and moving average (MA) processes may be present which would call for the application of an autoregressive moving average model (ARMA). To simultaneously examine k time series variables, the multivariate form of ARMA or Box–Jenkins analysis would be used. Pfeifer and Deutsch (1980), Kendall and Ord (1991) and Cressie (1993) suggest adding spatial dimensions to the model. To do this the time series of the study variable, measured at different locations, can be treated as a set of variables in a multivariate time series analysis. Cressie (1993) provides details on how ARMA (Box–

Jenkins) models can be generalized to include spatial location. There are space–time autoregressive (STAR) models and space–time autoregressive moving average (STARMA) models. The STARMA model uses an observation Z(si, t) taken at spatial location si and time t, and defines Z(t)⬅[Z(si; t), %, Z(sn;t)]⬘. The STARMA model takes the form (Cressie 1993, p. 449):

冘冘 p

Z(t)⫽

lk

k⫽0j⫽0

冘冘 q

ml

xkjWkjZ(t⫺k)⫺

fljVlje(t⫺1)⫹e(t)

l⫽0j⫽0

where Wkj and Vlj are given weight matrices, lk is the extent of spatial lagging on the autoregressive component, ml is the extent of spatial lagging on the moving average component, the residuals are given by e(t)⬅[e(si;t), …, ⑀(sn;t)]⬘ and xkj is the autoregressive parameter to be estimated while flj is the moving average parameter to be estimated. It should be noted that this represents the general form of the model and that some spatial or temporal autoregressive or moving average terms might not be found in the final model, because they were not statistically significant in this situation. Indeed, one might anticipate that simpler STAR (space– time autoregressive) models might be adequate in this situation. In particular, a simple first order space–time autoregressive model was initially thought to be most appropriate for these data, because the time series is relatively short, and observed beach flux and shoreline positional changes did not appear to exhibit any pseudocyclical movements associated with more complex spatial or temporal processes (see Tables 1 and 2, the spatial and temporal correlograms). The STAR models used in this study took the form: Ys,t=ys⫺1,tYs⫺1,t±ys,t⫺1Ys,t⫺1±ys⫺1,t⫺1Ys⫺1,t⫺1±e where the y terms represent the estimated autoregressive parameters and e represents the error or residual term.

5. Application and results When applying the Box–Jenkins modelling approach it should be noted that the fewer the number of parameters that have to be estimated, the simpler and more parsimonious the time series model. After the specific model parameter values are determined, the Box–Jenkins method provides statistical tests for verifying the appropriateness of the chosen model (see Box and Jenkins, 1976; Hoff, 1983; Pankratz, 1983). To determine the best fitting model for the beach data and for the shoreline data this paper considers the spatial series to be unidimensional, and that the dominant movement of sediment is unidirectional from north to south. In essence, the processes that produce change at profile 59A (distance = zero) should significantly influence the observed response at the other profiles. This influence decreases away from 59A starting at profile 160 which is 100 m

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Table 1 Spatial–temporal analysis of beach net sediment flux (BSF) patternsa Observed lag 1 2 3 4 5 6 7 8

Temporal ACF autocorrelation 0.174* ⫺0.053 ⫺0.062 0.024 0.038 0.081 ⫺0.021 ⫺0.121

S.E.

Observed lag

Spatial ACF autocorrelation

S.E.

0.063 0.064 0.065 0.065 0.065 0.065 0.065 0.065

1 2 3 4

0.273* 0.217* 0.102 0.028

0.075 0.077 0.082 0.097

S.E.

Box–Ljung Q

Prob.

0.070 0.078 0.081 0.085

0.211 3.109 3.189 3.216

0.646 0.205 0.363 0.522

Residuals from the model Observed lag 1 2 3 4 5 6 7 8

Temporal ACF autocorrelation 0.122 ⫺0.088 ⫺0.037 ⫺0.008 0.067 0.063 ⫺0.015 0.067

S.E.

Box-Ljung Q

Prob.

Observed Lag

0.063 0.063 0.064 0.064 0.064 0.064 0.065 0.065

0.130 2.271 2.649 2.667 3.906 5.001 5.063 6.303

0.718 0.132 0.266 0.446 0.418 0.416 0.536 0.504

1 2 3 4

Spatial ACF autocorrelation ⫺0.028 0.105 ⫺0.009 0.010

a Best Fit Model: (BSF)s,t=0.27 (BSF)s⫺1,t+0.15 (BSF)s,t⫺1±e (where s=profile position; e=residual). R2 is between observations and STAR model=0.106. *=significant at 0.05 level. ACF⬅Autocorrelation Function; SE⬅Standard Error. Box–Ljung Q=n(n+2)⌺ra2 (n⫺k)⫺1 which is distributed as c2 with degrees of freedom k–p where n is the number of observations, k is the number of lags used, p is the number of parameters and ra is an autocorrelation coefficient.

south of 59A. Field investigations showed that at any given time (Ti) interval, the profile (P) changes at distances one to eight exhibited seasonal variations. Hence, the data collected at the eight profile transects shown in Fig. 1 for thirty-four different time intervals (semiannually from 1978–1994) are represented in the modified Box–Jenkins model as: T1:P1,P2,P3,P4,P5,P6,P7,P8 T2:P1,P2,P3,P4,P5,P6,P7,P8 . . . T34:P1,P2,P3,P4,P5,P6,P7,P8 The Box–Jenkins results are obtained with the STATISTICA software time series modules (StatSoft Inc., 1995). Best fit models are obtained through a process that involves examination of a number of tentative model identifications and parameter estimations. Based on the autocorrelation functions depicted in Table 1 showing peak spatial and temporal autocorrelations at lag=1 which decline as the lag number increases, a space–time autoregressive (STAR) model was thought to best describe spatial–temporal variations in the beach net

sediment flux (BSF) data. The results presented in Table 1 indicate that the BSF data are fitted to a STAR model which accounts for 10.6% (R2=0.106) of observed spatial–temporal variation of BSF values. This model takes the form: BSFs,t=0.27 (BSF)s⫺1,t+0.15 (BSF)s,t⫺1±e, where s is the profile position (based on distance from 59A), t is the time period number, and e is the residual or error term. This model differs from the general form of the model presented earlier in that the first order space–time coefficient was 0.00, and no significant moving average terms were encountered. Although the model only accounts for 10.6 percent of the spatial–temporal variation of beach net sediment flux (BSF) patterns, the results are still statistically significant and one would expect a very large random element to be present in the data. Table 1 indicates that no significant autocorrelation exists in the pattern of residuals from the model as can be observed by the non-significant Box–Ljung Q values. Q is distributed as c2 with degrees of freedom equal to the total number of lags used minus the number of parameters and their associated probability values (any probability greater than 0.05 is considered non-significant). The model results highlight the fact that localized sediment flow processes have some control on beach net sediment fluxes. These processes operate through space, and it is evident that there are interrelated spatial patterns of erosion and accretion through time. One of the major

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Table 2 Spatial–temporal analyis of net shoreline positional changes (NSC) patternsa Observed lag

1 2 3 4 5 6 7 8

Temporal S.E. ACF autocorrelation 0.495* 0.152* ⫺0.004 ⫺0.050 ⫺0.062 ⫺0.036 ⫺0.109 ⫺0.109

0.063 0.076 0.077 0.077 0.078 0.078 0.078 0.078

Observed Lag

Spatial ACF autocorrelation

S.E.

1 2 3 4

0.542* 0.279* 0.074 0.065

0.093 0.115 0.120 0.123

Residuals from the model Observed lag

1 2 3 4 5 6 7 8

Temporal S.E. ACF autocorrelation ⫺0.015 ⫺0.025 ⫺0.022 ⫺0.020 ⫺0.026 0.058 ⫺0.092 0.001

0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063

Box–Ljung Q

Prob.

Observed Lag

Spatial ACF autocorrelation

S.E.

Box–Ljung Q

Prob.

0.058 0.221 0.348 0.454 0.634 1.525 3.777 3.792

0.809 0.638 0.840 0.928 0.959 0.910 0.707 0.803

1 2 3 4

⫺0.028 ⫺0.062 ⫺0.109 0.033

0.068 0.071 0.073 0.077

0.068 1.088 3.897 4.173

0.646 0.296 0.142 0.249

a Best Fit Model: (NSC)s,t=0.48 (NSC)s⫺1,t+0.42 (NSC)s,t⫺1⫺0.20 (NSC)s⫺1,t⫺1±e (where s = profile position; e = residual). R2 is between observations and STAR model =0.391; *= significant at the 0.05 level; ACF⬅Autocorrelation Function; SE⬅Standard Error. Box–Ljung Q=n(n+2)⌺ra2 (n⫺k)⫺1 which is distributed as c2 with degrees of freedom k⫺p where n is the number of observations, k is the number of lags used, p is the number of parameters and ra is an autocorrelation coefficient.

implications derived from this model is that beach flux levels in adjacent sites are interdependent, and represent the cumulative spatial effects of short-term sediment flow processes. These processes operating at the Northeast Beach are variations in longshore drift as well as persistent localized sediment circulation cells. The spatial and temporal variations explained by the model for the BSF data can be visualized on a space– time metamap. The space–time metamap is a variant of the Minkowski space–time diagram (see Mills, 1994, p. 52–56) which is used to summarize the salient patterns of variables which change through time and space. The profile locations, survey time and BSF data were input as variables into the STATISTICA software package (StatSoft Inc., 1995) to produce a two dimensional contour plot. A cubic spline subroutine was used to smooth the contour lines presented in the metamap (Fig. 2). The overall spatial–temporal trends presented in the metamap are indicative of the erosion and accretion patterns which have occurred through time and space. Low rates of erosion in the first 100 m south of 59A give way to modest accretion, especially for areas beyond the 260 m mark for the months before May 1985. Another aggradational pattern can be seen in the first 260 m of the beach for the months after May 1987. The broad pattern of erosion,

Fig. 2. Space–time metamap of beach net sediment flux (BSF), Northeast Beach, 1978–1994. (This time space metamap treats the spatial– temporal patterns of beach net sediment flux (BSF) as existing in a time–space continuum and one axis reflects the spatial dimension parallel to the baseline while the other axis represents the temporal dimension. It should be noted that the data have been smoothed using a cubic spline so some of the random discontinuities have been filtered out.)

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observed in the central part of the metamap, developed between May 1985 and November 1986. This erosion is attributed to record high water levels which caused artificial structures to fail and a large embayment to develop. This promoted a reduction in the normal north to south movement of sediment. To determine which model best fits the net shoreline positional change (NSC) data the same set of procedures utilized for the BSF data are followed. Based on the autocorrelation functions depicted in Table 2 showing significant spatial and temporal autocorrelations at lag=1 which rapidly decline as the lag number increases, it was thought that the best fitting model is a space–time autoregressive (STAR) model. Therefore, a STAR model of the form: (NSC)s,t=0.48(NSC)s⫺1,t+0.42(NSC)s,t⫺1 ⫺0.20(NSC)s⫺1,t⫺1±e was fitted to the data. Here, s is the profile position, t is the time period number, and e is the residual or error term. Overall, Table 2 shows that the STAR model accounts for 39% (R2=0.391) of the spatial–temporal variation of shoreline retreat levels. This relationship is stronger than that observed for beach net sediment flux, because shoreline positional change is more sensitive to water level changes than beach net sediment flux, and it is also less sensitive to random movements associated with longshore drift, and the movements associated with nearshore circulation cells. Table 2 indicates that the residuals from the STAR model of shoreline change do not have any significant spatial or temporal autocorrelation effects. This is shown by the non-significant Box–Ljung Q values and their associated probability values (probability values greater than 0.05 are considered non-significant). Hence, this model can be confidently accepted. The observed spatial and temporal patterns can be associated with dynamic temporal processes which cumulatively impact particular spatial locations. Shoreline positional change and total sediment flux tend to be more sensitive to short-term fluctuations in sediment dynamics between spatial locations. The existence of a temporal term in the STAR model can be explained by the fact that a large scale shoreline positional change can develop quite rapidly, but the change can persist for an extended period of time. The adjustments of the shoreline toward its original state will occur at a relatively slow rate. At the Northeast Beach the tetrapods which were emplaced to protect the shoreline now constitute an artificial reef, and these artificial structures have mitigated against the process of beach and shoreline restoration. The NSC results from the STAR model can also be highlighted on a space–time metamap (Fig. 3). These results show that there is a pattern of initial shoreline advance giving way to retreat, followed by a pattern of reduced shoreline advance through time. Along the spatial dimension, the intensity of shoreline retreat extends for about 260 m starting at profile 59A. Shoreline advance is restricted to the first eighteen months of the

305

Fig. 3. Space–time metamap of net shoreline positional change (NSC), Northeast Beach, 1978–1994. (This time space metamap treats the spatial–temporal patterns of net shoreline positional change (NSC) as existing in a time–space continuum and one axis reflects the spatial dimension parallel to the baseline while the other axis represents the temporal dimension. It should be noted that the data have been smoothed using a cubic spline so some of the random discontinuities have been filtered out.)

monitoring period (May 1978 to November 1979). This advance can be attributed to relatively low lake levels. Between May 1982 and May 1991 shoreline retreat dominates the Northeast Beach with pronounced retreat occurring between zero and 260 m. The distinctive ‘thumbprint’ configuration which represents rapid shoreline retreat is located wholly within that part of the beach which is served by the tetrapods. This reach of massive shoreline retreat may be associated with the combined effects of the armor stone breakwater, tetrapod failure and unusually high lake levels. Since 1977, Lake Erie water levels have been rising at a rate of 0.06 m per annum. Thornburn (1986) reported that Lake Erie water levels exceeded record levels by 0.06 to 0.37 m from June through October 1986. As the water levels peaked in 1986 the tetrapods became submerged and waves started to break over the sunken tetrapods. A large embayment developed behind the line formerly occupied by the concrete tetrapod system. With only relatively moderate rates of shoreline retreat beyond the 260 m mark it can be claimed that the beach areas not protected by the tetrapod system experienced less erosion than the areas with artificial structures.

6. Conclusions Temporal and spatial variations in the beach and shoreline environment are produced by a combination of highly complex interactions between the influences of anthropogenic activities, and the initial morphodynamic state of the beach governed by natural processes, among them changes in water levels, tides, waves, currents, winds and sediment supply. Understanding these changes are important because “in addition to being fun-

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damental to understanding the morphodynamics of beaches, the spatial and temporal behavior of the beach profile has direct application in coastal engineering projects involving beach nourishment and in the siting of coastal structures” (Larson and Kraus, 1994, p. 76). The findings of this study emphasize the fact that an assessment of beach and shoreline stability requires examining a long time series of data. Crowell et al. (1993) recommended using accurate data spanning the longest possible period of record in order to dampen decadallength trends while Eliot and Clarke (1989) claimed that at least ten years of continuous beach and shoreline data are needed before long-term trends can be established with confidence. Admittedly, difficulties exist in accurately predicting beach and shoreline changes (see Everts and Gibson, 1983). Uncertainties arise when attempting to link shoreline responses to the secular, cyclic, and aperiodic processes operating in the nearshore system (see for example, Morton, 1991; Fenster and Dolan, 1993). However, the acquisition of longer and more extensive beach survey measurements can enhance beach management decisions (Eliot and Clarke, 1989). Collected data must be modelled for spatial and temporal trends. The procedures employed in this study are appropriate for application elsewhere because they provide results which will enable a shoreline manager to ascertain whether localized stochastic spatial processes operate in the beach and nearshore system. The Box–Jenkins modelling results highlight the influence of temporal stochastic processes on the long range behavior of beach and shoreline variations. For instance, temporal processes, such as high water levels can have the effect of shifting a beach away from a state of dynamic equilibrium. Since many natural beaches can attain dynamic equilibrium conditions (e.g. Lakhan and Trenhaile, 1989; Larson, 1991; Pilkey et al., 1993) it is vital to utilize space–time modelling procedures in order to detect shifts in phase states which occur through time and space. The knowledge that a beach alternates between erosional and depositional states, together with the fact that short-term erosional trends are part of the longer dynamic equilibrium conditions of the coastal system, will enable coastal engineers and coastal resource managers to develop better solutions for beach and shoreline protection.

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