What is the fractal dimension?

This section of the manual will explain what the fractal dimension is and how RivEX calculates it.

What is the fractal dimension?

• The fractal dimension (fd) is a value that can be calculated for many natural phenomena. When applied to rivers the value of fd  lies between 1 and 2. A value of 2 would indicate that the river network is so branching that it fills the whole of 2-dimensional space.

• A higher fd value indicates a more complex river pattern and could help explain species distribution or hydrological responses.

• A popular method for calculating fd is to carry out "box counting" over a range of different cell sizes.  A grid is placed over the river network and the number of cells intersecting the river is counted.  This process is repeated using a smaller cell size and is continued until it becomes impractical to calculate.  Once these values are calculated they are converted to Log10 and plotted.  The slope of the best fit line for the plot of Log10 of cell count against Log10 of 1/ cell size is the fd.  The sequence of images below show this process diagrammatically.

• The cell size is divided into 1 before taking the Log so the slope becomes a positive value.

How does RivEX calculate the fractal dimension?

• RivEX calculates the fd for each catchment rather than the whole network.  A catchment is defined as all polylines flowing to a river mouth node.  You need to have run your data through RivEX to have calculated the catchment ID numbers before you attempt to calculate the fd.

• RivEX will select all polylines for a catchment and compute the extent of this selection. The extent is sometime referred to as the minimum bounding box.  In the image below you can see 3 catchments, note that the extent of two of the catchments overlap.

• RivEX then generates a series of grids with the cell sizes you have chosen.  It takes the catchment's extent and divides the width and height by the cell size to calculate the number of rows and columns the grid will consist of.  The grids are extended by 1 row and column to ensure that as the cell size increases it actually covers the extent of the catchment.

• To explain why an extra row/column is added, imagine a catchment generated an extent of exactly 1Km2 and you the user had chosen cell sizes in 250m increments. 1Km can be divided exactly by 250m and 500m but not 750m.  You need at least 2 cells of 750m width to cover the 1Km length of the catchment's width.

• The origin of all the grids (bottom left hand corner) are anchored exactly at the origin of the extent of the catchment and not the whole network.

• The selected polylines are then used to do a spatial intersect of the grid and the number of polygons selected are recorded. This process is repeated for all grids generated.

• RivEX gives you a choice in where to send the data, dbase or Excel.  If you choose dbase the counts and their logarithmic values are stored.  If you choose Excel then the same values are stored but RivEX also calls the in-built Excel function Slope to compute the fractal dimension value and this is stored as a value in the worksheet.