[1] The ready availability of satellite and other images of the Earth together with increasing societal interest in coastal processes and morphology make it desirable to have a consistent way of defining and mapping shorelines from images. The obvious choice, the land-water interface, is inadequate because it includes the back sides of islands and the edges of river and tidal channels that are not directly exposed to open water. On complex coasts, inclusion of these portions of the land-water interface would significantly affect quantitative measures of shoreline geometry. Here we propose a method to map shorelines consistently while accounting for ambiguity. The Opening AngleMethod (OAM) uses a visibility criterion to define the shoreline. First a region of unambiguous open water is defined. Next, the shoreline is defined as the locus of points for which the sum of viewing angles extending to open water, unobstructed by land, exceeds a specified critical value. Shorelines defined by the OAM follow the land-water interface where the shoreline is unambiguous, and in more complex cases OAM shorelines are consistent with the idea that the shoreline is that part of the land-water interface that is directly exposed to open water.

[2] Shorelines are one of the most important geographical features of our planet. They mark the boundary between terrestrial and marine (or lacustrine) domains, defining a fundamental process transition. Shoreline change is important on both human and geological time scales [Morton et al., 2004; Kim et al., 2006]. Shoreline morphology is a sensitive indicator of dominant formative processes (e.g., rivers, waves, and tides) [Galloway, 1975]. Finally, the shoreline is interesting as a geometric shape; it is no accident that the original example of a natural fractal was the coast of Britain [Mandelbrot, 1967].

[3] Quantifying the structure and evolution of plan-view geomorphic patterns such as river channels has been a productive approach to the study of Earth-surface processes [e.g., Nikora and Sapozhnikov, 1993; Rodríguez-Iturbe and Rinaldo, 1997; Sapozhnikov and Foufoula-Georgiou, 1996,1997; Syvitski, 2006]. However, while researchers have long recognized that shoreline shape and evolution reflect key sedimentary processes [Galloway, 1975; Boyd et al., 1992], quantifying this relationship remains elusive. The possibility of using measures of shoreline shape and change to make quantitative inferences about processes shaping the coast raises the question of precisely what the shoreline is.

[4] Many researchers have addressed issues arising in high-precision shoreline mapping of wave-dominated beach and barrier island settings (e.g., Figure 1a) [Boak and Turner, 2005; Morton et al., 2004]. However defining the shoreline on complex coasts (e.g., Figures 1b and 1c) is more difficult. In particular, the distinction between shoreline and other land-water boundaries, such as channel banks or sheltered parts of islands, is not clear. Many studies rely on qualitative characterizations of shoreline shape [Orton and Reading, 1993; Harris et al., 2002], or ignore complex coasts altogether (e.g., Morton et al. [2004] address only beach and barrier island shorelines). Here we present a new method of defining shoreline that provides a reasonable definition of shoreline on complex coasts and reproduces the common definition of the shoreline in simple cases. We call it the Opening-Angle Method (OAM).

2. Defining the Shoreline

[5] The shoreline separates a standing body of water (lake or sea) from land. The obvious definition of the shoreline is the land-water interface (LWI) [Boak and Turner, 2005]. This is straightforward, but where coastal morphology is complex the LWI includes at least two classes of points that are not directly exposed to open water: channel banks (e.g., Figure 1c) and the sheltered back sides of islands (e.g., Figure 1a). On some complex coasts, notably river deltas and salt marshes, these areas constitute a large fraction of the land-water interface.

[6] A common alternative is to define the shoreline as a particular elevation contour of the land (or water) surface (e.g., mean sea level) [Robertson et al., 2004]. However even if sufficient elevation data were available to track a contour around complex channelized coasts, this definition has the same problem as the LWI; that is, it includes points not directly exposed to open-water processes. More involved methods might define the shoreline using, for example, salinity, or some measure of received wave energy. These methods require too much information to be applied routinely to coastal images (e.g., time history of directional wave spectra together with a means of determining which waves are morphodynamically significant). They also fail completely in important special cases, for example, lakes in the case of a salinity-based definition, and shorelines in still water in the case of a wave-based definition.

[7] Overhead imagery is readily available and simple to acquire (including historical satellite and aerial data sets), and it is typically easy to distinguish land from water based on pixel value, particularly in infrared imagery. In such images the LWI is well defined, and image-based shoreline mapping has been very successful for simple coasts [Boak and Turner, 2005]. Based on these considerations, we propose a new image-based shoreline mapping method that is based on the LWI but can be applied to complex coasts. The Opening-Angle Method builds on the idea that the shoreline is that part of the LWI that is directly exposed to open-water. The quantitative measure we use to approximate the degree of exposure at a point is the angle Θ of open water the point can see, or “opening angle” (Figure 2). A point on the LWI that does not have a direct line of sight to open water (point A) is not part of the shoreline, but a beach that sees 180° of open ocean clearly is (point B). The appeal of the OAM is that it is a direct measure of exposure to open water.

3. Opening Angle Method

3.1. Theory

[8] The OAM is defined as follows. We begin with a binary map (Figure 3a), which consists of 1) a set of points P that forms a contiguous and bounded 2D domain (typically a rectangle), and 2) a subset WP of water points. From this, we extract the set of land points, L = P − W, and the set of points on the land-water interface, I (the boundary of L). In general L and W are each a contiguous 2D region (or finite collection of regions, if islands or lakes are present), while I is a continuous curve (or finite collection of curves).

[9] Next we define a contiguous subset W_{O}W as open water. The OAM can be used with any well defined open-water set W_{O}; the idea is to include only that part of the image that is unambiguously open water. An objective and consistent way to define W_{O} is to use the concept of the convex hull [Cormen et al., 2001]. The convex hull of the land, L_{C}, is the smallest convex polygon that contains L (outlined by red curve in Figure 3b). We define open water as all points in W that are outside the convex hull of the land, W_{O} = W − L_{C} (white region in Figure 3b).

[10] The opening angle Θ[x] for a point x in P is then defined as the sum of all swath angles θ that originate at x, extend to open water, and are bounded by rays tangent to the LWI (Figure 2). The opening angle Θ is defined for every point in the domain P (Figure 3c). However, all land points have an opening angle of zero (Θ[L] = 0°), because any open-water view must pass through the LWI. All open-water points have an opening angle of at least 180° (Θ[W_{O}] ≥ 180°), due to the definition of the convex hull. The points of interest are then water points inside the convex hull, W − W_{O}, and points in I.

[11] The simple OAM presented here is scale-independent, aside from image size and resolution. However for coasts with strong large-scale concavity (e.g., narrow embayments or closed water bodies) the convex-hull method will not perform well in defining the open-water set W_{O}, and it may be preferable to include a length scale in the definition of Θ. In these cases W_{O} can be defined using a generalization of the convex hull (e.g., alpha-shapes [Edelsbrunner et al., 1983]), or visibility can be defined using a maximum blocking distance (a horizon radius beyond which land is unseen). Either approach would allow extension of the OAM to cases where open water is between the tips of large-scale coastal embayments or is surrounded by land.

3.2. OAM Shorelines

[12] The OAM can produce two types of shoreline map, depending on the needs of the user. A continuous OAM shoreline classifies an image into terrestrial and marine components, and is defined by contouring the opening angle map Θ[x] for a given angle, Θ_{c}. Example continuous OAM shorelines are shown in Figures 3c and 4. Continuous OAM shorelines are useful for evaluating changes in coastal land area, for example, due to wetland loss or delta growth. While a continuous OAM shoreline typically crosses water, traversing mouths of channels or inlets, a discontinuous OAM shoreline rejects these points, defining the shoreline as the set of segments in I with Θ ≥ Θ_{c}. For any Θ_{c}, the discontinuous OAM shoreline is given by the intersection of the continuous OAM shoreline with the LWI. Discontinuous OAM shorelines are useful for evaluating how fragmented a particular coast is.

[13] The OAM provides a family of potential shorelines depending on the choice of cutoff angle Θ_{c}. Figure 3c shows that the largest and most protruding elements of the LWI are always part of the shoreline, while more sheltered and/or convoluted segments are the first to be lost as Θ_{c} is increased. This ambiguity is a fundamental part of the shoreline-definition problem that cannot be circumvented; the OAM provides a method for evaluating it quantitatively and objectively. However we have found that Θ_{c} = 45° produces a reasonable shoreline for a variety of coastal configurations, and we propose this as a standard value for defining shorelines using the OAM. Figures 3c and 4 show the 45° shoreline on three example coastlines.

[14] Useful information can also be extracted from analysis of the dependence of OAM shorelines on Θ_{c}. The shoreline on simply shaped coasts varies only slightly with changes in Θ_{c}. For complex coasts the shape and length of the shorelines vary greatly with changes in Θ_{c}, because the extent to which an OAM shoreline bends into channels depends upon the minimum view of the ocean that it is allowed to have. Thus, the change in shoreline length with Θ_{c} provides a convenient measure of shoreline complexity (Figure 4c).

4. Example Applications

[15] To demonstrate the use of the OAM we consider three coasts of varying complexity. Optimal application of the OAM requires images where the coast in question is contained entirely within the chosen image without truncation and is not significantly concave at large scales. The examples used here satisfy both of these requirements. The Outer Banks (Figure 1a) are a group of sandy wave-dominated barrier islands backed by lagoons. The Selenga Delta (Figure 1b) is a river-dominated delta with abundant distributary channels. The Bengal Delta (Figure 1c) is a large delta with strong tidal influence. To apply the OAM to discrete images, we use a fast approximation that gives results close to the exact OAM (see Methods). For each coast we compute shoreline length as a function of Θ_{c} (Figure 4c), and measure shoreline ambiguity as the fractional length increase between small (30°) and large (120°) Θ_{c} continuous OAM shorelines. Full resolution images and OAM maps for these coasts are available as auxiliary material.

[16] The Outer Banks is a smooth coast and the shoreline is unambiguous. This can be seen in the Θ map (Figure 4a), which shows that shorelines for the range of opening angles tightly track the LWI. Consequently, the length of shoreline does not change significantly with Θ_{c}, resulting in a relatively low shoreline ambiguity of 5%.

[17] On the Selenga Delta, feathered distributary extensions and standing water (marsh) on a variety of scales make it difficult to identify an unambiguous shoreline; in particular the unmodified LWI would certainly be unsatisfactory. This ambiguity is apparent in the Θ map (Figure 4b), which shows that, in contrast to the Outer Banks, here shoreline morphology changes significantly with Θ_{c}. This results in a much larger ambiguity of 23%. In spite of this the 45° OAM shoreline (white curve) tracks the shape of the delta well and is quite reasonable.

[18] The Bengal Delta has tidal and fluvial channels extending far inland; their mouths cover a wide range of scales, making it particularly difficult to map a consistent shoreline. Here the OAM inscribes arcs into the mouths consistently while leaving the headlands unambiguous (Figure 3d), i.e., different Θ_{c} shorelines diverge smoothly in channel mouths but converge on headlands. Due to its wide channels, this shoreline has an extremely large ambiguity of 81%.

5. Closing Comments

[19] Other researchers have noted the inherent ambiguity in measuring shorelines [Mandelbrot, 1967]. However the ambiguity identified by the Opening Angle Method is distinct from the well-known ambiguity associated with measurement scale. In our initial research, we explored wavelet-based approaches akin to the “Mandelbrot-ruler” method. We found that while ambiguity associated with river-mouths decreased at large ruler sizes L, these large-scale shorelines introduce spurious ambiguity on well-defined stretches of coast (e.g., headlands). As shown in Figures 3 and 4, regardless of the particular Θ_{c} threshold chosen, OAM shorelines follow the LWI on unambiguous stretches. Thus, an OAM shoreline provides a consistent and physically reasonable basis for fractal or other analysis of shoreline geometry. For a particular coast, “L-families” and “Θ-families” of shorelines (based on ruler length and opening angle respectively) can give complementary insights into morphology and coastal processes.

Appendix A:: Methods

[20] To use the OAM for image-based shoreline mapping we must adapt the theoretical description to apply to discrete data. Our definitions based on point sets are straightforward to translate into discrete forms using piecewise-linear polygons and curves, and the OAM could be rigorously implemented using techniques of computational geometry [Cormen et al., 2001]. For simplicity, here we implement an approximate OAM using point sets made up of pixel centroids x_{i}.

[21] In this approach, the input is the discrete set of image pixels, P = {x_{i}}, and the discrete set of land pixels, L, in the form of a discrete wet map (Figure 3a). We use false-color LandSAT Geocover images (Figure 1) where L can be easily made by thresholding the green band (NIR light) of the image subtracted from the blue band (green light). We pre-process L by filling lakes (contiguous sets of water pixels surrounded by land). Next we compute the discrete LWI as the set of land pixels I which have at least one neighboring water pixel. We then compute the convex hull of land pixels, L_{C}, using standard techniques [Cormen et al., 2001].

[22] The query set Q is the set of pixels where we compute Θ. For discontinuous OAM applications Q = I, while for continuous mapping Q = W − W_{O} (Figure 3b). An ocean view is blocked when a ray originating at a given pixel intersects a land pixel. In this discrete framework, we only consider blocking by land pixels that are either in I or on the edge of the image, which together define the test set T. For computational efficiency we only consider opening angles up to 180°, hence we can set Θ[W_{O}] = 180°, and since we know Θ[L] = 0°, we can exclude land and open ocean from Q.

[23] Given the query and test sets, we compute the discrete opening angle map, Θ[Q], using the following (schematic) algorithm: 1 For x_{i} in Q 2 For x_{j} in T, compute θ_{j} = angle[x_{i}, x_{j}] 3 Sort θ_{j} into descending order 4 Compute Δθ_{j} = θ_{j} − θ_{j+1} 5 Sort Δθ_{j} into descending order 6 Compute Θ[x_{i}] = ΣΔθ_{k}, k = 1, …, pIn this algorithm, for each query pixel we compute only the rays θ_{j} whose views are blocked by land (step 2). We then infer the angles of unobstructed views Δθ_{j} from gaps between these blocked views (step 4; Here we also account for ray-pairs which straddle 360°).

[24] Because T is not a continuous curve, query pixels can “see through” the discrete LWI. Most of these spurious views are between neighboring LWI pixels, and at this stage we remove these views. We minimize the impact of the remaining spurious views by considering only the p largest view angles (Line 6), where p <<∣T∣. (In this work we use p = 3, which allows for multiple views between islands and works quite well.) While it is not an issue on many coasts, this approximation returns some spuriously high opening angles if the coast has thin spits or islands (approaching one pixel in width). In this case our approximation creates spurious shoreline on the back side of islands. We circumvent this problem by first thresholding the Θ map at the critical angle Θ_{c} and then applying a small-windowed Gaussian blur (no larger than the width of the barrier) to the resulting binary image. A contour of the blurred image will then return a shoreline that closely approximates the exact OAM shoreline. This method was used in all examples in the paper.

Acknowledgments

[25] JBS is grateful to the Oberlin College January Term program, which allowed him to get started in this research. We thank Ravi Janardan for discussing the alpha-shapes method with us and Andrew Ashton and Brandon McElroy for thoughtful and very useful reviews. This work was supported by the STC program of the National Science Foundation via the National Center for Earth-surface Dynamics (EAR-0120914).