CE322 Basic Hydrology
Horton's Laws - Example
Jorge A. Ramirez
 
 

Obtain estimates of the Bifurcation Ratio, RB, the Length Ratio, RL, and the Area Ratio, RA, 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, Dd, and d) the average length of overland flow, Lo.

Order, w

Number of Streams

Average Length (km)

Average Area (km2)

1

60

2

5

2

13

5

12

3

9

13

40

4

4

20

110

5

1

55

330


 

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

(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, RL.

(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, RA.

(3)

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

  1. Taking logarithms of each of the above equations leads to:

(4)
 
 

(5)
 
 

(6)

These equations are linear in w. Thus, estimates of the RB, RL, and RA, 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, RB = 2.551635.

B) Law of Stream Lengths and Length Ratio:

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

C) Law of Stream Areas and Area Ratio:

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

D) The total length of streams can be calculated as:

Using the above equation leads to LT = 437 km.

E) Drainage density:

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

F) Average length of overland flow:

Thus, Lo = 0.377574 km = 377.74 m