## DESCRIPTION

r.broscoe.sh Calculates ADTDTM 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.
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=CAt

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).

Hw = Hw+1 = Hw+2 = ...

where Hw 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 (the suggested ADTDTM test and t-test), 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 t-test (according to the Tarboton approach) and the ADTDTM test for each threshold value.
For the ADTDTM test the parameter Pvalue is taken. The right threshold should be the first value just greather than the choosen statistical significance. Here a statistical significance valure of 0.05 is proposed.
An output graphs is generated as PDF in the working folder.

## EXAMPLE

An example on Chiascio stream (Umbria, Italy):
The command syntax:
```  r.broscoe.sh dem=dem_abt 'thresholds=2000 3000 4000 5000' xcoor=2320378.547 ycoor=4779694.770 lt=3 result=broscoe_chiascio
```
The results are an "output.csv":
```threshold	n1	n2	Mean 1			Mean >1			diff			sd 1			sd >1			TrMean 1		Tr Mean >1		diff			Test t eq var		Perm Test eq var	Perm Test noeq var	Perm Test diff TrMean
2000		127	47	59.3228346456693	64.1063829787234	-4.78354833305411	71.8789637294643	62.2363077459624	52.2869565217391	58.3953488372093	-6.10839231547018	0.687063371319418	0.663366336633663	0.712871287128713	0.534653465346535
3000		87	32	67.9885057471264	63.78125		4.20725574712644	79.5933698087265	68.0449255651095	60.1139240506329	57.1666666666667	2.94725738396625	0.791240399171673	0.801980198019802	0.801980198019802	0.693069306930693
4000		74	21	72.0945945945946	51.6190476190476	20.475546975547		100.947724016344	42.6104167903533	57.6764705882353	49.8421052631579	7.8343653250774		0.368259756306302	0.405940594059406	0.198019801980198	0.623762376237624
5000		60	16	76.4833333333333	61.0625			15.4208333333333	108.929224676854	52.2193690118906	59.0555555555556	61.0625			-2.00694444444444	0.585538501927895	0.683168316831683	0.623762376237624	0.920792079207921
```

..and an "output.pdf" file of the graphics where threshold values are natural (left) and logaritmic (right):

## 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.

v.strahler
r.strahler.sh

## REFERENCES

C. Cencetti et alii, (submitted to Computers & Geosciences). A more efficient statistical test for drainage networks delineation using GIS and the theory of MEAN STREAM DROP.

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.

## AUTHORS

Pierluigi De Rosa, Ivan Marchesini and Annalisa Minelli, Univ. Perugia.

Last changed: \$Date\$