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## fractal dimension

Started by Matteo Convertino, August 09, 2010, 09:12:28 AM

#### Matteo Convertino

Dear everyone.
In sea-level rise predictions SLAMM gives as output also the fractal dimension. (actually two fractal dimensions)
Is that the fractal dimension of the coastline ? or something else?
How is that determined?

#### Matteo Convertino

#1
so anyone knows exactly how this fractal dimension is calculated or what specifically is.
because i potentially find a correlation with with this variable and something else in my coupled model.
thanks

M

#### Jonathan S. Clough

#2
Greetings.  Back from two weeks out of the office so my apologies about the delay.  This calculation is designed to capture the marsh to water interface.  This dates back to work done in the early 1990s in which the marsh-to-water interface was found to correlate to shrimp catch in the Gulf of Mexico and so model predictions about this interface were then used to predict shrimp.  It has rarely been used in recent analyses.

There are two such calculations within the code, with different assumptions about the marsh-to-water interface.  The first calculates the interface between regularly flooded irregularly flooded and mangrove and open water, the second the interface between regularly-flooded marsh and open water (labeled as "S.M only")

References are The Science of Fractal Images, pages 59-62 especially, http://www.amazon.com/Science-Fractal-Images-Heinz-Otto-Peitgen/dp/0387966080.

Here is some background on the code design:
QuoteThe code is saving the minimum and maximum x and y coordinates where the water to marsh interface is found.  It is also saving the number of boxes in which the water to marsh interface is found (NBox).  The maximum length (LMax) is calculated using the maximum of the length or width in which interfaces are found. Finally fractal dimension is calculated as Log(NBox) / Log(LMax)

From page 60 of the Fractal Images Book,
Dimension is Log N / Log (1/r) where r is the ruler length, but N can be considered to be "number of squares containing a piece of the object."
Therefore, this function is assuming that LMAX = 1/ruler length.
I see where this is expanded on in page 61:  NBOX = LMAX ^ D which simplifies to the equation for D in the code.

Let me know if you have further questions.  -- J

#### Matteo Convertino

#3
Ok, so it`s box counting, thanks a lot!

- Matt

#### Jonathan S. Clough

#4
Well, to be precise, an accounting of the "box to length" relationship.  A marsh-to-water interface that is a straight line will have a low dimension and an interface that is very complex will have a higher fractal dimension.  This article is moderately informative -- http://en.wikipedia.org/wiki/Fractal_dimension

#### Matteo Convertino

#5
Jonathan, when you spoke about the two fractal dimensions, you meant it is calculated between the ``mangrove class and open water'' and between the ``salt marsh class and open water''. what about all the other classes ? because i can look also e.g. the interface between the ``ocean beach class and water'', between the estuarine class and open water'' and so  on..........

#### Matteo Convertino

#6
well considering only the salt marsh the Df is lower but in time it has the same trend of the other.

#### Jonathan S. Clough

#7
No there are only two fractal dimensions calculated with an eye towards correlation with shrimp catch

The regularly flooded marsh -- open water interface and

The marsh and mangrove -- open water interface.  "Marsh and mangrove" is defined as regularly flooded marsh plus irreg. flooded marsh plus mangrove.

-- Jonathan

#### Matteo Convertino

#8
Hi J.

so do you expect that the fractal dimension calculated for different interfaces e.g. water-ocean beach class , water-estuarine beach class etc... can produce different fractal dimensions? .... probably yes however I do not expect huge variations from the fractal dimensions already provided.
(Also because the range of the fractal dimension is small by itself)

best,

Matteo