SOURCE RECONSTRUCTION FOR A COASTAL EVOLUTION EQUATION
D E Reeve
School of Civil Engineering, University of Nottingham, NG7 2RD, UK.
Tel: +44 (0)115 951 4648; Fax: +44 (0)115 951 3898; Email: dominic.reeve@nottingham.ac.uk
M Spivack
DAMTP, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK.
Tel: +44 (0)1223 337861; Fax: +44 (0) 1223 337918; Email: ms100@damtp.cam.ac.uk
Abstract
The numerical prediction of seabed evolution over periods of time useful for
strategic management is in its infancy. This is due both to a scarcity of appropriate data and
a gap in our understanding of the physical processes important in governing the long-term
variations in coastal evolution. Recently developed inverse methods offer an opportunity of
improving our understanding of the patterns of long-term sediment movement and their
variation over time.
Introduction
In many coastal regions the seabed is composed of sediments that become mobile
under the action of moderate waves and tidal currents. Changes in large-scale morphology
of the seabed may have a profound effect on sea defences, navigation and shoreline
management. Developing a means of forecasting the future evolution of the morphology of
the seabed is thus of significant practical importance.
The prediction of seabed evolution over periods of time useful for strategic
management is in its infancy. This is due both to a scarcity of appropriate data and a gap in
our understanding of the physical processes important in governing the long term (ie
decadal) variations in coastal evolution.
Recently developed inverse methods offer an opportunity of improving our
understanding of the patterns of long term sediment movement and their variation over
time. An inverse method for analysing longshore coastal changes, Reeve & Spivack (1994),
has been used for analysing longshore coastal changes as part of several strategic Shoreline
Management Plans in the United Kingdom. The inversion technique described in this paper
is 3-d (2 space and 1 time dimension), and can be applied to features such as sandbanks,
spits and bars.
The new inversion technique has been applied to an area covering the offshore
sandbanks at Gt Yarmouth on the East Coast of the UK. The fluctuations in the patterns of
inferred sediment supply and removal over the period of 150 years provide a picture of the
changes in the long-term sediment transport regime.
Theoretical Considerations
Diffusion equations have been used in the coastal engineering literature to describe,
individually, long-shore and cross-shore sediment movement, Stive et al. (1991), Larsen et
al. (1997). The diffusion equation governing the long-shore transport of sand on a beach
was derived by Pelnard-Considere (1953) on the basis of theoretical considerations and
physical model experiments. This model is also termed a 'one-line' model as it predicts the
position of a single depth contour. Niedoroda et al. (1995) have proposed a diffusion
equation with a source term to describe changes in cross-shore beach profiles.
The obvious extension to two dimensions is again to use a diffusion-type equation
to describe long-term changes in the level of the seabed surface, Spivack & Reeve (1999).
This form of equation can also be derived from the dynamical equations governing fluid
flow, with appropriate assumptions, and may be justified as being physically reasonable, in
that wave and tidal action is effective at smoothing or eroding high relief features.
However, there are some features such as sandbanks that are also maintained by a
combination of waves and tides. The inclusion of a general source/sink term provides a
mechanism to maintain seabed gradients without describing specific processes in the
equation explicitly. Here we take the equation governing changes in the level of the seabed
surface, h(x,y,t), to be:
h
 
h 
 
h 

  F ( x, y , t )

K y ( y)
 K x ( x)


t
x 
x  y 
y 

(1)
where x , y and t are cross-shore direction, longshore direction and time respectively. Kx(x)
and Ky(y) have the role of diffusion coefficients for the cross-shore and longshore
directions respectively, and F is a source/sink function. By writing the diffusion coefficients
as the sum of a constant reference value and a spatially varying component, Equation 1 may
be recast in the form:
h
 2h
 2h
 Kx

K
 G( x, y, t )
y
t
x 2
y 2
(2)
where Kx and Ky are the reference values of the diffusion coefficients in the cross-shore and
longshore directions respectively. The additional terms involving the spatially varying
components of the diffusion coefficients have been incorporated into the modified source
term G(x,y,t)
Numerical Solution
It may be shown, Spivack & Reeve (1999), that if the time variation in G is
neglected the solution of Equation 2 over any time step  can be written formally as
h(t   )  [(exp( D )  1) D 1G  exp( D )]( h(t ))
(3)
whereD denotes the horizontal Laplacian, (D2). By expanding the second term in a
Taylor series Equation 3 can be simplified to
h(t   )  [ exp(D / 2)G  exp(D)]( h( ))
(4)
The terms involving D are pure diffusion-type terms and can be expressed using
two-dimensional Fourier transforms. Specifically, we have


exp(D / 2)h( x, y, t )  F 1 exp( (k 2  l 2 ) / 2 F (h( x, y, t ))
(5)
where F is the two-dimensional Fourier transform with respect to x and y, k and l are the
corresponding transform variables and F-1 is the inverse transform. A similar expression
holds for the second term.
The inverse problem consists of determining G(x,y) from Equation 4. The inverse of
the expression acting on G in Equation 4 is given by exp(-D/2). Multiplying Equation 5
through by this term and collecting up the exponential terms on the right hand side of he
equation gives:
1
G ( x, y )  [exp( D / 2)h(t   )  exp(D / 2)h(t )]

(6)
Given the bathymetry at two different times, t and t+, we can evaluate the terms on
the right hand side of Equation 6 via discrete Fourier transform to yield the source term.
The inversion requires values of the diffusion coefficients to be prescribed. Here, values of
103 m2/year were used which is at the bottom of the range of values typically used in oneline models of sandy beaches. Note that it is not possible to resolve the details of G more
finely than the points at which H is known.
Computational Results
The inversion technique has been applied to an area covering the offshore
sandbanks at Great Yarmouth on the East Coast of the UK. The area considered is
approximately 12km wide and 35km long. Historical data were abstracted from sixteen
bathymetric charts for dates between 1846 and 1992, for an area that extends from
Winterton in the north to Benacre in the south, see Figure 1.
Key features of the sand bank system are:








Holm Sand, almost 3km due east of Lowestoft;
Corton Sand, ~5km due east of Gorleston-on-Sea;
Middle and North Scroby Sands lying ~2km offshore and running almost
parallel to the mainland between Gt Yarmouth and Newport;
South, Middle and North Cross Sands running almost parallel to the Scroby
Sands but approximately another 3km further offshore;
Caister Shoal which runs parallel to the mainland between California and
Winterton, approximately 1km offshore.
Holm Channel;
Barley Pickle (a deep channel);
Caister Road (also a channel).
The dates of the charts used were 1846, 1864, 1875, 1886, 1896, 1905, 1916, 1922,
1934, 1946, 1954, 1962, 1974, 1982, 1987 and 1992. Maps earlier than this are available.
However, maps and charts earlier than 1800 are not usually reliable, in that they are rarely
based on an adequate ground-triangulation control, and often were simply sketch surveys,
Carr (1962). The earlier charts in particular may contain data from several surveys over a
period of months or more. The data were digitised and then reduced to a common
projection and datum (OSGB38 Ordnance Survey National Grid). The reduced data were
then used to create a TIN (Triangular Irregular Network) model for each survey date. The
TIN models were then used to interpolate to a set of digital terrain models, (DTMs),
covering the area of the sandbanks at a resolution of 100m. The data on which the charts
were drawn had a mean density of ~30 points per km2, corresponding to a mean resolution
of approximately 180m.
Contour maps showing the digital seabed models for 1982 and 1992 are shown in Figures 2
and 3 respectively. Some of the major changes in the sandbanks between 1982 and 1992
are: the growth of Holm Sand, Corton Sand and the southeastern section of Middle Scroby;
the extension of the channel between Middle Scroby and Corton Sand further south and
east.
Figure 4 shows the results of the inversion process for the 1982 and 1992 maps.
Positive/negative values correspond to a supply/removal of sediment through dynamical
processes other than those described by diffusion over the period 1982-1992.
Comment on features on result….
Include more solutions?
Conclusions
Inverse techniques can provide a useful adjunct to more traditional prediction
models and may lead to improved forecasting methods through the improved understanding
of the underlying physical processes that they can bring. We have derived and implemented
a method for reconstructing a source term in a simplified governing equation for seabed
evolution where time variation is slow. Data measurements are required on a regular grid of
points at intervals over time. Although these data requirements are fairly demanding, such
measurements are available in some regions of interest to coastal engineers and managers.
The results shown here demonstrate the feasibility of the approach.
Acknowledgements
The authors would like to acknowledge the useful discussions with Dr Li Bin of
Halcrow Group Ltd. This work was partially funded by the UK Government's Ministry of
Agriculture, Fisheries and Food under Contract No. FD1008, and was commenced while
the first author, (DER), was a member of staff with Halcrow Group Ltd.
References
Carr, A.P., 1962. Cartographic record and historical accuracy, Geography, 47 (2), p135144.
Clayton, K.M., McCave, I.N. & Vincent, C.E., 1983. The establishment of a sand budget
for the East Anglian coast and its implications for coast stability, Shoreline Protection
Conference, Thomas Telford, p91-96.
Halcrow, 1991. The Future of Shoreline Management, Conference Papers, Publ. National
Rivers Authority Anglian Region & Sir William Halcrow & Partners Ltd., pp116.
Larson, M., Hanson, H. & Kraus, N.C., 1997. Analytical solutions of one-line model for
shoreline change near coastal structures, J. Wtrwy., Port, Coast., and Oc. Engrg., ASCE,
July/Aug. p180-191.
Niedoroda, A.W., Reed, C.W., Swift, D.J.P., Arato, H. & Hoyanagi, K., 1995. Modelling
shore-normal large-scale coastal evolution, Marine Geology, 126, p181-199.
Pelnard-Considere, R., 1956. Essai de theorie de l'evolution des formes de rivage en plages
de sables et de galets, Societe Hydrotechnique de France, Iveme Journees de
L'Hydraulique, Question III, Rapport 1, 74-1-10.
Reeve, D.E. & Spivack, M., 1994. Determination of a source term in a linear diffusion
equation, Inverse Problems, 10, p1335-1344.
Robinson, A.H.W., 1960. Ebb-flood channel systems in sandy bays and estuaries,
Geography, 45, p183-199.
Robinson, A.H.W., 1966. Residual currents in relation to shoreline evolution of the East
Anglian coast, Marine Geology, 4, p57-84.
Spivack, M. & Reeve, D.E., 1999. Source reconstruction in a coastal evolution equation,
Submitted to J.Phys.Comp.
KEY WORDS
Coast
Evolution
Equation
Inverse
Reconstruction
Source
Morphology
Long-term
Management
Sandbank
Numerical