transect locations. Various methods of determining shoreline rates of change have been described by Dolan
et al. (1991) and Kumar et al. (2010), which are widely
considered the definitive works on the subject. All
methods used for calculating shoreline rates of change
involve measuring the differences between shoreline positions through time. Rates of shoreline change are
expressed in terms of distance of change per year.
Methods, such as end-point rate, average of rates, linear
regression, jackknife, and average of eras rates, are
adapted in estimation of shoreline changes.
End-point rate (EPR)
End-point rate can be calculated by dividing from the two
end points – the earliest and latest positions – the distance
of shoreline movement by the time elapsed between the
earliest and latest measurements at each transect. The
future shoreline position for a given date can be estimated
using the rate and intercept (Fenster et al., 1993):
Y ¼ mX þ B
ð1Þ
where Y denotes the shoreline position, X the date, B the
intercept, and m the rate of shoreline movement. Given
shoreline data sets, numbered in ascending order by date,
the EPR intercept is
B EPR ¼ Y n À m EPR Â X n
ð2Þ
Average of rates (AOR)
This method involves calculating separate end-point rates
for all combinations of shorelines when more than two are
available and can be extended to incorporate the accuracy
of the shoreline position data and the magnitude of the rate
of change. Foster and Savage (1989) developed an equation to determine if any given EPR meets a minimum time
criterion (T min ):
T min ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E 1
ð Þ
2 þ E 2
ð Þ
2
q
R 1
ð3Þ
where E 1 and E 2 are the measurement errors in the first and
second point, respectively, and R 1 is the EPR of the longest time span for a particular transect.
The advantages of using AOR are that all the EPRs that
survive the minimum time span equation are used and it
allows calculation of the time-dependent variance from
the average of rates. The two primary disadvantages of
using AOR to compute long-term trend are as follows:
(1) there is no computational norm for modeling the
minimum time span equation, and (2) the results are
sensitive to the choice of values selected to represent the
measurement errors (Dolan et al., 1991). Foster and
Savage (1989) do not recommend AOR as a general
computational method, but it can be used as a method of
verification in combination with EPR and LR.
Linear regression (LR)
A linear regression rate-of-change statistic can be determined by fitting a least squares regression line to all shoreline points for a particular transect. LR is the most reliable
method to predict future shoreline positions and their associated confidence intervals, if measurement errors and
a linear trend of erosion were the only determining factors
over the longest possible period of shoreline position
(Crowell et al., 1997; Douglas and Crowell, 2000). Linear
regression can reveal if an association exists, and in particular (via the r value) what fraction of the variance of the
dependent variable (shoreline position) is attributable to
the independent variable (time). This method uses all the
available data from many data sets to find a line, which
has the overall minimum of the squared distance to the
known shoreline.
To calculate the rate of change and to predict the future
shoreline position using linear regression, the oldest
shoreline position is chosen as a baseline or zero (0) position to measure the amount of shoreline shift (Maiti and
Bhattacharya, 2009; Kumar et al., 2010). With reference
to this baseline, progradation of the shoreline is considered as a positive value, and recession is considered as
a negative value. The change in shoreline position rate is
calculated by the linear regression equation y ¼ a + bt,
where y is the shoreline shift during the year t, with
y ¼ 0 for t ¼ X. The regression coefficient (b) represents
shoreline-change rate, and the correlation coefficient (r) is
a measure of goodness of fit of the equation to the present
data. Based on the number of samples, the statistical
significance is to be considered at the 80 % level of
confidence (if number of samples is small) and 95 % confidence level in the case of large numbers of samples as
suggested by Allan et al., (2003).
Jackknife
The jackknife method is used as an iterative linear regression that calculates a linear regression fit to shoreline data
points with all possible combinations of shoreline points,
leaving out one point in each iteration. The slopes of the
linear regression lines are averaged to yield the jackknife
rate. The advantages of the jackknife are similar to linear
regression; the jackknife is also less influenced by outliers
of data clusters. The main disadvantage of the jackknife is
a lack of increased statistical value given the typically
small numbers of shoreline data points used to derive
a shoreline rate of change. The statistical utility of the
jackknife is best realized with an order of magnitude of
(more) data points.
Summary
Shoreline, a part of coastal land in contact with estuarine
or ocean waters, undergoes various morphodynamical
changes. The shorelines migrate landward or seaward
depending on changing sea level or uplift or subsidence
of coastal regions. The cyclic and noncyclic processes
change the position of shorelines over time scales, from
SHORELINE CHANGES
599
et al. (1991) and Kumar et al. (2010), which are widely
considered the definitive works on the subject. All
methods used for calculating shoreline rates of change
involve measuring the differences between shoreline positions through time. Rates of shoreline change are
expressed in terms of distance of change per year.
Methods, such as end-point rate, average of rates, linear
regression, jackknife, and average of eras rates, are
adapted in estimation of shoreline changes.
End-point rate (EPR)
End-point rate can be calculated by dividing from the two
end points – the earliest and latest positions – the distance
of shoreline movement by the time elapsed between the
earliest and latest measurements at each transect. The
future shoreline position for a given date can be estimated
using the rate and intercept (Fenster et al., 1993):
Y ¼ mX þ B
ð1Þ
where Y denotes the shoreline position, X the date, B the
intercept, and m the rate of shoreline movement. Given
shoreline data sets, numbered in ascending order by date,
the EPR intercept is
B EPR ¼ Y n À m EPR Â X n
ð2Þ
Average of rates (AOR)
This method involves calculating separate end-point rates
for all combinations of shorelines when more than two are
available and can be extended to incorporate the accuracy
of the shoreline position data and the magnitude of the rate
of change. Foster and Savage (1989) developed an equation to determine if any given EPR meets a minimum time
criterion (T min ):
T min ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E 1
ð Þ
2 þ E 2
ð Þ
2
q
R 1
ð3Þ
where E 1 and E 2 are the measurement errors in the first and
second point, respectively, and R 1 is the EPR of the longest time span for a particular transect.
The advantages of using AOR are that all the EPRs that
survive the minimum time span equation are used and it
allows calculation of the time-dependent variance from
the average of rates. The two primary disadvantages of
using AOR to compute long-term trend are as follows:
(1) there is no computational norm for modeling the
minimum time span equation, and (2) the results are
sensitive to the choice of values selected to represent the
measurement errors (Dolan et al., 1991). Foster and
Savage (1989) do not recommend AOR as a general
computational method, but it can be used as a method of
verification in combination with EPR and LR.
Linear regression (LR)
A linear regression rate-of-change statistic can be determined by fitting a least squares regression line to all shoreline points for a particular transect. LR is the most reliable
method to predict future shoreline positions and their associated confidence intervals, if measurement errors and
a linear trend of erosion were the only determining factors
over the longest possible period of shoreline position
(Crowell et al., 1997; Douglas and Crowell, 2000). Linear
regression can reveal if an association exists, and in particular (via the r value) what fraction of the variance of the
dependent variable (shoreline position) is attributable to
the independent variable (time). This method uses all the
available data from many data sets to find a line, which
has the overall minimum of the squared distance to the
known shoreline.
To calculate the rate of change and to predict the future
shoreline position using linear regression, the oldest
shoreline position is chosen as a baseline or zero (0) position to measure the amount of shoreline shift (Maiti and
Bhattacharya, 2009; Kumar et al., 2010). With reference
to this baseline, progradation of the shoreline is considered as a positive value, and recession is considered as
a negative value. The change in shoreline position rate is
calculated by the linear regression equation y ¼ a + bt,
where y is the shoreline shift during the year t, with
y ¼ 0 for t ¼ X. The regression coefficient (b) represents
shoreline-change rate, and the correlation coefficient (r) is
a measure of goodness of fit of the equation to the present
data. Based on the number of samples, the statistical
significance is to be considered at the 80 % level of
confidence (if number of samples is small) and 95 % confidence level in the case of large numbers of samples as
suggested by Allan et al., (2003).
Jackknife
The jackknife method is used as an iterative linear regression that calculates a linear regression fit to shoreline data
points with all possible combinations of shoreline points,
leaving out one point in each iteration. The slopes of the
linear regression lines are averaged to yield the jackknife
rate. The advantages of the jackknife are similar to linear
regression; the jackknife is also less influenced by outliers
of data clusters. The main disadvantage of the jackknife is
a lack of increased statistical value given the typically
small numbers of shoreline data points used to derive
a shoreline rate of change. The statistical utility of the
jackknife is best realized with an order of magnitude of
(more) data points.
Summary
Shoreline, a part of coastal land in contact with estuarine
or ocean waters, undergoes various morphodynamical
changes. The shorelines migrate landward or seaward
depending on changing sea level or uplift or subsidence
of coastal regions. The cyclic and noncyclic processes
change the position of shorelines over time scales, from
SHORELINE CHANGES
599