Mapping coastlines

SUMMARY: Fractals can be used to help map coastlines.

A map is an infographic representing an area. Maps use symbols to represent objects or scale renderings of spatial features. The science of mapmaking is called “cartography.” The mapping of coastlines is important for navigation and for determining the boundaries of territorial waters, which are measured as fixed distances from coastlines. Coastline cartography presents special mathematics because of connections with several actively developing branches of mathematics, including fractal theory.

Traditional Mapmaking Mathematics and Analytical Cartography

Several mathematical features of maps have been used for centuries. Orientation is the correspondence of the map’s coordinate system with directions of the terrain. When three-dimensional objects are depicted in two-dimensional media in the process called “projection”such as maps of Earth on papersome areas are necessarily distorted. Ratios are used to map objects to scale, including the systematic changes in the ratios in different parts of two-dimensional maps using projections.

With the increasing use of computers in cartography, several new areas of modeling and computation expertise have appeared over the last few decades. These new, mathematics-rich cartography areas include computer-based geographic information systems, interpolation, and photogrammetry. Collectively, these areas of expertise are called “analytic cartography.”

Types of data in analytic cartography include numerical data, such as elevation values, images or photographs, and attribute data, like tags identifying features near particular coordinates. All data are dynamically linked and manipulated in a geographic information system; for example, a projection map can be generated from a series of aerial photos, rotated and zoomed. In contrast, paper maps do not allow dynamic data connection and are static, which limits the possibility for mathematical modeling and experimentation with variables. Geographic information systems may also include remote sensing data; for example, displaying changes in coastlines in real time as tides change.

Analytic Cartography and Coastline Changes

Because coastlines change a lot compared to other map features, from tides and floods, analytic cartography that allows for rapid analysis of real-time data is especially valuable in mapping coastlines. Using data from previous events and mathematical models within geographic information systems, cartographers can simulate floods, tsunamis, or effects of rising water levels from global warming on existing coastlines. The same software can be used to predict effects of terrain modification projects over time.

Modeling coastline changes is more complex than simply mapping higher or lower water levels onto the existing coast elevation data. The models also have to take into account erosion, deposits of matter by rivers and rainfall runoff, changes in river basins, and other systemic factors.

Fractal Dimension and the Coastline Paradox

A fractal is a self-similar structure that looks the same at all zoom levels. Coastlines, while not perfectly fractal (not having infinite number of levels), exhibit enough fractal features to make some mathematics of fractals applicable. The famous 1967 paper by Benoît Mandelbrot, “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” started this line of thought, though the term “fractal” appeared later.

An important feature of a coast is its fractal dimensiona measure of how long the coast is compared to the area it occupies. Because the area has two coordinate dimensions and the length has one, theoretically, a curve filling a unit of area can have infinite length. Fractal dimension is a way to compare different coasts, from straight coastlines that have the fractal dimension of 1 to increasingly complex, space-filling coastlines that have higher fractal dimensions between 1 and 2. In Mandelbrot’s paper, the relatively smooth coast of South Africa has the fractal dimension of 1.02 and the highly irregularlong for its areacoast of Britain has the fractal dimension of 1.25.

The length of the coast and its fractal dimension depend on the units of measure. Because smaller units allow the cartographer to capture more detail of the coastline, measuring with smaller units produces higher total length. This is definitely not true about measuring straight lines, and thus it is called the “coastline paradox.”

Randomness and Pattern

Perfectly self-similar fractals created by mathematical models have limited applicability to coastline mapping because real coasts are irregular. Therefore, some mathematical models include the element of randomness in creation of factors and use statistical methods to compute fractal dimensions. For example, one method for generating random fractals is called “random midpoint displacement,” produced by using the following cyclic algorithm repeatedly:

• Step 1: Start with a straight line.
• Step 2: Displace the midpoint randomly, perpendicular to itself, by the distance within the given ratio to its length.
• Step 3: Apply Steps 1 and 2 to the segments resulting from the previous steps.

A similar method can be applied to generating elevation of areas. In this case, the algorithm starts with a rectangle, displaces its midpoint, and then is applied to the four rectangles formed by the lines parallel to the original rectangle’s sides and crossing at the midpoint.

Because these methods are computationally intensive, as the number of computations at each step grows exponentially with the number of cycles, their development coincides with increases in computing power. In addition to mathematical modeling of existing coasts, these methods are used to generate fictional terrain for computer games, virtual worlds, and digital artworks.

Coast-Mapping Satellites

Several government and private projects connect real-time satellite data to specialized coastline geographic information systems. This connection provides either real-time or within-minutes data for ship navigation charts, environmental hazards (like oil spills in harbors), and natural disaster data (like tracking tsunamis).

Satellite mapping has to use methods beyond optical imagery because data have to come during the night as well as in cloudy conditions. Coast-mapping satellites use radar sensors that do not depend on light. These sensors measure changes in reflected radar pulses. Rougher surfaces reflect differently from water, allowing for relatively precise mapping of the coastline.

Artificial Intelligence for Coast-Mapping

In the 2020s, coast mapping became a critical function as global climate change altered the shape of the earth's shorelines. As projections showed steadily rising sea levels of 0.25 to 0.3m over the next three decades. Accurate coastal mapping was essential for civil planning efforts. This was particularly true for major metropolitan areas that would have to deal with these significant terrestrial changes. At least 32 major coastal cities in the United States were at risk of future inundation of areas currently inhabited by human populations. In addition to living areas, impacts to ecosystems and vital infrastructure were also projected.

In the 2020s, the employment of new technologies such as Artificial Intelligence (AI) allowed for more accurate modeling of future coastal changes. AI was adept at predicting sea level characteristics, such as projected temperature variations over long time periods. AI was also used to project sea surges, variations in wind and pressure, and factor in topographical characteristics of coastlines.

Bibliography

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