CE520 Physical Hydrology

Jorge A. Ramirez

Associate Professor

 

Horton Laws - Example

 

 

Obtain estimates of the Bifurcation Ratio, R_B, the Length Ratio, R_L, and the Area Ratio, R_A, using the data tabulated below. Also, for the same data, a) determine the area of the basin, b) the total length of streams, c) the drainage density, D_d, and d) the average length of overland flow, L_o.

 

 

         The Horton law of stream numbers states that there exists a geometric relationship between the number of streams of a given order N_w and the corresponding order, w. The parameter of this geometric relationship is the Bifurcation Ratio, R_B.

                                                                                                         (1)

         The Horton law of stream lengths states that there exists a geometric relationship between the average length of streams of a given order and the corresponding order, w. The parameter of this relationship is the so-called Length Ratio, R_L.

                                                                                                     (2)

         The Horton law of stream areas states that there exists a geometric relationship between the average area drained by streams of a given order and the corresponding order w. The parameter of this relationship is the so-called Area Ratio, R_A.

                                                                                                   (3)

In the equations above, W is the order of the basin, and the over-bar indicates the average value of the corresponding variable.

 

a)             Taking logarithms of each of the above equations leads to:

 

                                            (4)

 

                                 (5)

 

                               (6)

These equations are linear in w. Thus, estimates of the R_B, R_L, and R_A, can be obtained by linear regression of:

,

,

,

respectively. Denoting by m the slopes of the corresponding fits, the above estimates are obtained as:

For the problem at hand:

 

Order	Number of Streams	Average Length	Average Area	log(N)	log(L)	log(A)
1	60	2	5	1.778151	0.30103	0.69897
2	13	5	12	1.113943	0.69897	1.079181
3	9	13	40	0.954243	1.113943	1.60206
4	4	20	110	0.60206	1.30103	2.041393
5	1	55	330	0	1.740363	2.518514

A) Law of Stream Numbers and Bifurcation Ratio:

The linear regression analysis returns a slope m = -0.40682. Thus, R_B = 2.551635.

Law of Stream Lengths and Length Ratio:

The linear regression analysis returns a slope m = 0.348073. Thus, R_L = 2.228807.

 

Law of Stream Areas and Area Ratio:

The linear regression analysis returns a slope m = 0.46013. Thus, R_A = 2.884894.

 

The total length of streams can be calculated as:

Using the above equation leads to L_T = 437 km.

Drainage density:

Thus, D_d = (437 km)/(330 km²) = 1.3242 km^-1

Average length of overland flow:

Thus, L_o = 0.377574 km = 377.74 m