**CE322 Basic Hydrology**

**Horton's Laws - Example**

Jorge A. Ramirez

Obtain estimates of the Bifurcation Ratio, *R _{B}*, the Length Ratio,

Order, w |
Number of Streams |
Average Length ( |
Average Area ( |

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 *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.

- 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}*,

,

,

,

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 |

The linear regression analysis returns a slope *m* = -0.40682. Thus, *R _{B}* = 2.551635.

B) Law of Stream Lengths and Length Ratio:

The linear regression analysis returns a slope *m* = 0.348073. Thus, *R _{L}* = 2.228807.

C) Law of Stream Areas and Area Ratio:

The linear regression analysis returns a slope *m* = 0.46013. Thus, *R _{A}* = 2.884894.

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

Using the above equation leads to *L _{T}* = 437

E) Drainage density:

Thus, *D _{d}* = (437

F) Average length of overland flow:

Thus, *L _{o}* = 0.377574