GRASS GIS: r.broscoe.sh

NAME

r.broscoe.sh

DESCRIPTION

r.broscoe.sh Calculates waerden test and t test statistics for some values of threshold area on a single basin, according to A.J.Broscoe theory (1959).
The program uses some R commands for statistical analisys and graphic rapresentation. In particoular the R package "agricolae" is required.
The A.J.Broscoe theory is well known as the theory of the "Mean Stream Drop" and it says that, for the extraction by DEM of a stream network, exists a threshold value wich makes drop constant, and this is the right one extraction threshold.
By definig the drop (H) as:

H = S L

for streams in each Strahler order.
The drop is calculated as the quote difference by ending and starting point of a stream; S is the slope and L is the lenght of the same stream.
Using the Leopold and Miller relation (1964):

S=CA^t

where C and t are constat values; the area we are searching (A) is the lowest that gives S to find H costant for each Strahler order (w).

H_w = H_w+1 = H_w+2 = ...

where H_w is the mean of the drops related to the streams in the same Strahler order (w).
The area can be found by making some attempts for different area thresholds, doing some statistical tests (Van der Waerden test and linear regression), and choosing the right threshold from the output of the tests.

r.broscoe.sh takes in input the DEM, the threshold values on wich calculate statistics, the outlet coords of the basin you want to study; it returns a table (text file) with the output of the Van der Waerden test and linear regression (t test) for each threshold value.
For the Van der Waerden test the parameter Pvalue is taken. It has to be greater than the possible, it represents the possibility of success of the test (the Mean Stream Drop is the same for all Strahler orders).
For the linear regression the parameters t, Pr, R_squared_adj are taken. t is the t statistic value, Pr is the possibility of success of the t test, R_squared_adj measures the dispersion of data around the mean value (for each order) for given degrees of freedom.
Three graphics called "linear_regression", "waerden_test" and "all_tests" are also generated as PDF in the home folder.

Preferably let's take the threshold value wich gives Pvalue (or Pr) greater than 0.95, but is not granted that you can reach that result because it depends of the well-graduation (by Horton-Strahler) of the basin, its geomorphological maturity, so it is not rare that you have to take threshold where Pvalue is simlpy the greatest.
At the end of the calculation, at first Pvalue is examinated, then, only if Van der Waerden test gives no good results (low Pvalue), the linear regression output (Pr) is examinated; in fact the Van der Waerden test is preferred to linear regression because it allows you to consider the real dispersion of data around the mean: this makes you able to know the real significance of the probability (e.g. the significance is low for few data in the sample) considering an unique parameter.

EXAMPLE

An example on Menotre stream (Umbria, Italy):
The syntax:

  r.broscoe.sh dem=dtm20_regione@AB 'thresholds=400 600 800 1000 1200 1400 1800 2000' xcoor=2291350.34 ycoor=4765192.22 lt=4 result=menotre_txt

The results:

threshold	t	Pr	Radj	Pvalue
400	0.5713518	0.568486	-0.003798402	0.6085511
600	0.8791352	0.3810997	-0.001896266	0.2798474
800	1.053110	0.2948033	0.001067895	0.29454
1000	0.02578308	0.9794938	-0.01233737	0.8535388
1200	0.3985548	0.69147		-0.01234108	0.6340721
1400	-1.024254	0.3100425	0.0008457844	0.256408
1800	-0.6368832	0.5274277	-0.01309044	0.5764749
2000	-0.4003206	0.6908575	-0.01901582	0.814699

By the report and graphics, you can see that the Van der Werden test gives not-so-good results (Pvalue_max=0.85 for threshold=1000 cells) but, if you consider the linear regression output (Pr), you can see that for the same threshold value (1000 cells) Pr is 97%.
So the threshold=1000 cells is chosen. Moreover the program returns a set of vector map called "orderd_thresholdvalue" from wich you can extract the right one orderd-network (in this case the right one is "orderd_1000"), you can rename and use it as well as you want.

NOTES

The lt value requested in input is a parameter that prevents eventual errors in the DEM; it considers the presence of pits and represents the height difference lesserthan a drop is not considered as a drop but as a pit, and extracted from Mean Stream Drop analysis.

The program uses the module r.strahler, so it presents the same conditions about the selection of a threshold value range.

REFERENCES

NIST, (2006). Van Der Waerden.
URL: http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/vanderwa.htm

D. G. Tarboton and D. P. Ames, (2001). Advances in the mapping of flow networks from digital elevation data. World Water and Environment Resources Congress, presentation (2001).

J. J. Flint, (1974). Stream gradient as a function of order, magnitude, and discharge. Water Resources Research, vol.10, n.5, p.969-973.

NIST, (2006). Engineering statistical handbook: confidence limits for the mean.
URL: http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm

J. C. Davis, (1990). Statistics and Data Analysis in Geology. John Wiley \& Sons editors (New York, NY, USA).

A. J. Broscoe, (1959). Quantitative analysis of longitudinal stream profiles of small watersheds. Department of Geology, Columbia University, NY.

F. De Mendiburu, (2006). Statistical Procedures for Agricultural Research.
URL: http://rss.acs.unt.edu/Rdoc/library/agricolae/html/agricolae.package.html

AUTHORS

Ivan Marchesini and Annalisa Minelli, Univ. Perugia.

Last changed: $Date$