line the coast long is How ain? of Britcoastalgeotools.org/wp-content/uploads/Parrish.pdf · 15 20...

4/17/20154/17/20154/17/2015

How long is the coastline

of Britain?>> Ask Dan.

Mar 30 – Apr 02, 2015

By:

D. Michael Parrish, PhD, PE, CFM

© 2015 CRIO Group. All Rights Reserved. Download Permitted.

The title of my presentation is taken from a famous paper by Mandelbrot. The subtitle refers to a database of coastline lengths, currently under construction. A more subdued subtitle might have been “Ask DAN.” By the way, the words “coastline” and “shoreline” have the same meaning, and I will use them interchangeably.

Overview

Background

Calculating Coastline Length

Reporting Coastline Length

Questions / Discussion

© 2015 CRIO Group. Coastal GeoTools I P 1

My presentation has two focal points. They are proposed methods of 1) calculating and 2) reporting coastline length. Before I get to those, I’ll give some background.

BACKGROUND


First, I’d like to share a few examples of the utility of coastline length. How might coastline length be applied in a practical way?

Uses of Coastline Length

Map update planning:

“When allocating map

modernization funds, FEMA is

encouraged to prioritize as criteria

the number of... coastal miles

within the State...” among others

(111 HR 298, 2010; FEMA 2011a).


In 2010, the United States Congress said that when allocating map modernization funds, FEMA is encouraged to prioritize by the number of coastal miles. This is reflected in a 2011 report from FEMA to Congress.

Uses of Coastline Length

Estimating Erosion

“To calculate long-term coastal erosion rates in the

Laptev Sea and subsequent sediment release to

the Arctic Ocean, a precise assessment of shoreline

length is necessary” (Kruse et al. 2005).

Ecological Studies

Coastline length as one parameter quantifying

habitat (e.g., Messmer et al.) and fishing pressure

(e.g., Stewart et al. 2010).


Additionally, coastline length has been applied to estimating erosion as well as fishing pressure, which is related to the amount of fishing in a given area relative to the ability of the fish population to recover.

DEFINITION OF

“COASTLINE”


Coastline length seems to be an important quantity for certain problems. But, how do we define coastline length? There appears not to be a standard, or at least not a standard that is widely used. How coastline length is defined depends on who you ask.

“Coastline”

“Which Shoreline? The

choice comes down to the

best fit, a combination of:

Scale, Tidal datum,

Coverage, Format”.

(Goodwin et al. 2009)


I would like to take a step back for a moment and talk about what we mean when we use the word “coastline,” then move on to coastline length. A presentation given at Coastal GeoTools2009 by Goodwin, Miller, and Stein addresses this question. They boil it down to map scale; tidal datum or vertical reference; coverage, which is the horizontal extent being considered; and data format, for example vector, raster, or tabular data.

“Coastline”


In researching the definition of “coastline,” I took an informal survey of fifteen people. I asked them whether they considered the points shown on this drawing to be on the coastline.

“Coastline”

no

may…

yes

0

5

10

15

A

B

C

maybe

no

yes


Everyone agreed that point A—on the left—was on the coastline. But there was disagreement about the other two points. The disagreement seemed to be related to the definition of coastline. Some commented that Point B—near the bottom of the screen—looks like it’s on the bank of a river. Some said that the coastline is where the land meets the sea. To those people, it was obvious that all three points are on the coastline.

“Coastline”

Coastline of What?

Landmass

Include Rivers?

Land mass and body of water

Political Unit (Land-based)

Include (Smaller) Islands?

Include Rivers?

Examples:

Coastline of Britain

British coast of the

North Sea

United Kingdom


I believe the problem of defining which points are on the coastline is related to how the question is posed. For example, we could ask the question in Mandelbrot’s paper, “how long is the coastline of Britain?” A more specific question might be, “How long is the British coast of the North Sea?” In other words, we could specify the land mass only, or the land mass as well as a body of water. If only the landmass is specified, there is a question of whether to include rivers. One might also specify a political unit rather than a land mass. In this case, there is the additional question of whether to include islands, or which islands to include.

“Coastline”

Coastline of What?

Landmass

Include Rivers?

Land mass and body of water

Political Unit (Land-based)

Include (Smaller) Islands?

Include Rivers?

Proposed Standard

Exclude Rivers

Include only the named

body of water

Include mainland only,

Exclude Rivers


I would propose that if the inclusion of rivers is not specified, then they ought to be excluded from a calculation of coastline length. Similarly, if only a political unit is specified, islands ought to be excluded. If the political unit consists mainly of islands, such as the United Kingdom, only the large island should be included.

“Coastline”

Source:

The National Atlas of the United States of America.

General Reference", compiled by U.S. Geological Survey

2001, printed 2002

Denmark, Greenland: CIA (2014) Atlas


There are some interesting exceptions, but these point the need to be specific. How long is the coastline of Denmark? It depends on whether we include Greenland. How long is the coast of Hawaii? It depends on whether we mean the island or the state.

COASTLINE LOCATION

AND LENGTH


Now that I’ve proposed a way to clarify what we should mean when we say coastline, or coastline of a particular place, or the coastline of a particular body of water adjacent to a particular country, I will propose a method of locating coastlines and of calculating their lengths.

Coastline Location

Procedure

Select Horizontal Datum

Select Vertical Datum

Select DEM

Type

Element Shape

Orientation and Position

Source

Resolution

Proposed Standard

Administrative Datum

Tidal

MHW

Administrative (e.g. NAVD 88)

Select DEM

Raster

Square

Aligned to Horiz. Datum

Publicly Available

Grid size << Span


Before I move on, I’d like to explain the layout of this slide. I have two columns. On the left are the necessary steps, and on the right I have my proposed standard.

The first group of activities to undertake in locating the coastline is to select horizontal and vertical datums for the project, and to select a digital elevation model. I think the horizontal datum should match a projected administrative datum, such as a State Plane. I think the vertical datum should either be Mean High Water or an administrative Datum that is based on tides, such as NAVD 88. I don’t think datums lower than Mean High water should be used for a standard coastline, since these areas are inundated daily. I also do not think mean higher high water should be used for a standard definition of coastline, because some areas of the world do not have mean higher high water distinct from mean high water. I think the DEM should be in raster format, have square cells, be aligned with the horizontal datum, and be publicly available from a government or academic institution—preferably for little or no cost to the user. Finally, the cell size should be much smaller than the span. I will say more about the span in a minute.

Coastline Location

Procedure

Identify Land Cells

Proposed Standard

Land > Datum


Now that the digital elevation model and datums have been selected, the location of the coastline can be computed. Land cells are identified as those cells above the vertical datum. The other cells are Sea cells. Next, the boundary between Land and Sea is identified.

Coastline Location

Procedure

Identify Land /

Sea Boundary

Standard

DEM cell

edges shared

by Land and

Sea cells


This map represents the land / sea boundary near the River Thames…

The inset in the lower-left corner highlights some crossings, places where the land / sea boundary crosses itself. Crossings in the boundary indicate points where the boundary features are smaller than the cell size of the digital elevation model. There are also some smaller boundary segments that are completely disconnected from the others.

Coastline Location


Procedure

Identify

Coastline

Standard

Contiguous

Boundary

The next step is to identify a continuous boundary representing the coastline of each land mass under consideration. This map represents a contiguous coastline in red. You can make out the crossings in the land / sea boundary more clearly here, where the red an blue meet.

Coastline Location

Procedure

Identify Coast Points

Proposed Standard

DEM cell corners on

Coastline


Finally, points on the coastline are identified at the corners of the cells of the digital elevation model. I will call these coast points. Keep in mind these coast points do not have a reference to a specific body of water, only to a particular land mass.

Coastline Length

Procedure Proposed Standard

Mark Coastline Points

Using Dividers

* From fig. 2.9 in Luzadder (1952)

Calculate Coastline

Length


Once we have the coast points, we are ready to measure coastline length. In order to measure coastline length, I propose to use the method preferred by Mandelbrot, adapted to modern computers. He suggests walking along a mapped coastline using dividers. For those unfamiliar with mechanical drawing tools, dividers are similar to a compass, but with two metal points rather than one metal point and one pencil point.

Coastline Length


Walking along a mapped coastline with the dividers would look something like this.

Coastline Length


Coastline Length


Other methods exist, but they are not necessarily better. Mandelbrot’s method is straightforward and relatively easy to implement. Additionally, the literature on coastline measurement, and measurement of complex geometric figures in general seem to lead to his 1967 paper referenced in the title of this presentation. It does not bother me that this paper is almost 50 years old. Age alone does not invalidate a method. Newton’s laws of motion can still be used to calculate the positions of the planets, even though his laws of motion are over 300 years old.

For the computer scientists in the room, Mandelbrot’s method, as adapted here, can be applied without using floating point operations.

Coastline Length

Procedure Proposed Standard

Mark Coastline Points

Using Dividers

Use a raster approx.

of circle

Calculate Coastline

Length


In order to adapt the method to modern computers, I propose using a numerical template that represents an arc traced out by the divider. The numerical template is a raster that approximates that arc, like the circular pattern of squares shown here. I am calling the distance between the two legs of the divider the divider span. The divider span is also equivalent to the radius of the numerical template.

Coastline Length


If we were to mark the coast points of this map using the template, we would get…

Coastline Length


…something like this. The black dots on this map illustrate the marked coast points.

Coastline Length

Procedure

Apply Project-Specific

Definition of Coastline

Proposed Standard

Standard (for Landmass

Coastline, Unspecified Body of

Water)

Create one-sided Buffer on

“wet” side of Coastline

Buffer Width = Span

Eliminate Coastline Points

“outside” of Buffer


After walking around the coastline with the dividers, we have identified coast points that each represent a length of coastline proportional to the divider span. The next step in defining coastline length is to apply the definition of coastline in the applicable context. What are we calculating the coastline of? Are we including rivers or not? This might seem backwards, to locate the coasts points before applying the coastline definition. However, proceeding in the way I have described allows multiple users to apply the same set of coast points in different ways. Had we eliminated sections of coastline at the outset, a future user of the coast point data may not have all of the coast points needed. Maybe the first user wants to exclude rivers and the second user wants to include them.

Coastline Length

Seaward

Boundary

Of Buffer

River Thames


I suggested earlier that if we ask how long the coastline of a country is, we should exclude rivers. So, for the purposes of illustration, here is an example of elimination using a buffer, which is a band of uniform width surrounding the coastline. In this illustration, only those points near the seaward boundary of the buffer are included in the calculation of coastline length. Most of the River Thames is excluded.

REPORTING COASTLINE

LENGTH - EXAMPLES


Once coastline length has been calculated, how is it typically reported, and how should it be reported?

Coastal flood hazard analyses will be

performed… along 211.0 miles of

shoreline.

(FEMA 2011b)

.


Coastline length is often reported in a deceptively straightforward manner, as in this flood study report. Here we have a precision of tenths of a mile (2009 Coastal Scoping Report for Baldwin and Mobile Counties Alabama).


Miles of Coast: 2,344 Source:

http://coastalmanagement.noaa.gov


According to NOAA Ocean and Coastal Resource Management the Georgia coastline is about 2000 miles long.



The problem with reporting coastline length as simply as in the examples just shown is that coastline length depends on the divider span. This is similar to what happens to apparent detail as you look more closely at the coastline.


Divider Span:

2 km

Total Length:

9540 km


As you reduce the span…


Divider Span:

1 km

Total Length:

10,669 km


… more details are captured in the measurement process, and a greater length is measured. In theory, coastline lengths approach infinity as the divider span approaches zero. Only smooth curves have measured lengths that stabilize as divider span decreases…

log Length

log Divider Span

Theoretical Length

Smooth Curve

Great Britain (West Coast)*

Highly Complex Curve


* Mandelbrot (1967)


…as represented by the horizontal, red line on this plot. Here, I am representing divider span on the horizontal axis, with divider span increasing as one goes to the right, and decreasing as one goes to the left. On the vertical axis, I am representing coastline length, with length increasing as you go up. Coastline length would plot as a sloped line on this plot. For example, as we move left along the blue line, the divider span decreases, and coastline length increases. The slope of this line is consistent with Mandelbrot’s description of the west coast of Britain. The slope of the black line corresponds to a curve that is extremely complex.



The NOAA shoreline website has a one-paragraph description of a calculation of its official coastline length for the United States. This is an improvement over the previous examples.


Vučetić et al. (2006)

Scale

1 : 25 000

1 : 50 000

1 : 100 000

1 : 200 000

1 : 300 000

1 : 500 000

1 : 1 000 000

1 : 2 000 000

Coastline

Length (km)

130

123

114

103

99

91

77

65


In this Croatian paper, the coastline length of one of the islands of Croatia is measured on maps of several different scales. The authors report the computed lengths with the associated scale in tabular form and on a log-linear plot. I think this is one of the better ways to report coastline length.

REPORTING COASTLINE

LENGTH - PROPOSED


What, then, is my preferred method of reporting coastline length?


Richardson’s Model of Coastline Length (Mandelbrot 1967)

Factor, F

Fractional Dimension, 1 < D < 2

Divider Span, G


I think that the reporting of coastline length should reflect the fact that as one reduces the divider span, measured coastline length increases. This idea is summarized in a model that Mandelbrot attributes to Richardson. In that model, coastline length is affected by three parameters: the divider span, a constant factor; and a fractional dimension, which is related to the complexity of the coastline shape. The factor and the fractional dimension are fixed for a given coastline specified by two endpoints, and the divider span is arbitrary.


L = F G 1 – D

L: Coastline Length

F: Factor

D: Fractional Dimension, 1 < D < 2

G: Divider Span


The model can be expressed as shown here. One may estimate the factor and the fractional dimension after computing the coastline length using two distinct divider spans.


Simple Reporting (When Necessary):

L1 = 5041 km

L2 = 4422 km

½ L1 – ½ L2 = 309 km ≈ 500 km

Report as 4700 km


If the coastline length is to be reported with only some digits and a unit, I suggest that the unit be equal to the span and the number of significant digits reflect the effect of doubling the span.

log

L

eng

th

log Divider Span

Coastline Length


Complete Reporting

(When Feasible):

Factor, F

Fractional Dimension, D

Table

Graph

Divider

Span (km)

.25

.50

1.00

2.00

3.00

5.00

1 0.00

Coastline

Length (km)

130

123

114

103

99

91

77


Ideally, one would report the estimated factor and fractional dimension, and a table with several values of coastline length together with the corresponding divider spans. I also recommend presenting a plot of those tabulated values.


How long is the coastline of Britain?

Ask DAN

There is one more thing my abstract promises: an on-line tool for users to explore shoreline lengths.

DAN

DAN: Database of Coastline

Length Analyses

http://*.com

Contains tables describing

coastline length analyses

Source:

OST50: “Contains public sector information licensed

under the Open Government License v2.0.”


I am calling it D—A—N, Database of Coastline Length Analyses. It is currently in development. At a minimum, it will consist of a set of tables that fully describe coastline length for a number of land masses and bodies of water. I apologize to those who are disappointed by the fact that I have not directly answered the question posed in the title. However, those familiar with Mandelbrot’s original paper should not be surprised—he doesn’t answer the question either!

Thank You

D. Michael Parrish

Coastal Engineer

CRIO Group

[email protected]

CRIO Group

1801 South Australian Avenue, Suite 102West Palm Beach, FL 33409

Phone: (561) 253-9015www.criogroup.com


SELECTED REFERENCES

Hicks SD et al. (2000) Tide and Current Glossary. USDOC, NOAA, NOS, Center for Operational Oceanographic Products and Services. Silver Spring, Md.

Houghton Mifflin Harcourt Publishing Co. (ca. 2014) PARCC Test Prep question.

Kruse JE et al. (2005) Not just a matter of scale: measurement and comparison of coastline lengths derived from WVS, Landsat and Corona datasets for representative coast sections of the Lena Delta, 2nd International Alfred Wegener Symposium. Bremerhaven, Germany 31? Oct–2 Nov.

Luzadder WJ (1952) Fundamentals of Engineering Drawing, 3rd ed. Prentice-Hall; New York.

Mandelbrot BB (1967) How long is the coast of Britain? Statistical self similarity and fractional dimension. Science 156:636–638.

Messmer V et al. (2011) Habitat biodiversity as a determinant of fish community structure on coral reefs. Ecology 92:2285–2298. doi:10.1890/11-0037.1

Vučetić et al. (2006) The Effect of Map Scale on the Determination of the Coastline Length and the Area of Islands in the Adriatic Sea—the Example of the Island of Rab. http://hrcak.srce.hr/file/9670

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