La evolución de los sistemas pedagógicos en los conservatorios

The study of the geometric properties of mountains, also called orometry, attempts to describe through quantitative or numerical expressions. In the past, orometry was limited to the description of maximum and average altitudes, and establishing curves that represented the distribution of the surface of a basin as a function of altitude, but today, orometry and the larger field of morphometry involve a much larger range of indices.

Surface area

Because a watershed is by definition a delimited area which collects in its water-ways the precipitation on its surface, its discharge will be related to its surface area.

The surface area of a watershed can be measured by superimposing a grid drawn on transparent paper, using a planimeter, or preferably using digitizing techniques.

Shape

The shape of a watershed is an essential element because it influences directly the shape of the hydrograph of its outlet. For example, for the same rain event, a watershed with a long narrow shape will result in lower peak flows at the outlet, because it requires more time for the water to reach the outlet. Conversely, a fan-shaped watershed has a faster concentration time, resulting in a higher peak flow (if all other variables remain the same, Figure 3.9).

Various morphological indices can be used to characterize flows and to compare different watersheds. For example, the Gravelius shape index K_G is an index of compactness (or density) which is defined as the relation between the perimeter of a watershed and the perimeter of a circle with the same surface area. It is expressed by the following equation:

(3.5)

where K_G is the Gravelius shape index, A is the watershed area [km²], and P is the perimeter of the watershed [km].

This index is determined from a topographic map by measuring the perimeter and

the area of the watershed. The index is close to 1 for a watershed with a circular shape and is greater than 1 for a watershed that is elongated in shape (Figure 3.10).

Other indices include:

• Horton’s drainage density index (1932): defined as the relationship between the surface area of a watershed and the length of its principal waterways.

• Miller‘s circularity ratio (1953): defined as the relationship between the surface area of a watershed and that of a circle with the same perimeter length.

• Schumm’s elongation ratio (1956): represents the relationship between the diameter of a circle with a surface area equal to that of the watershed and the maximum length of the watershed.

Q/A

t Q/A

t t_c2 > t_c1

t_c1

Fig. 3.9 : Impact of the watershed shape on hydrograph.

K_G= 1.6 K_G= 1.3 K_G= 1.2 K_G= 1.1

Fig. 3.10 : Some catchments and their Gravelius shape indices.

Relief

The influence of the relief on flow is easy to understand because a number of hydro-meteorological parameters vary with altitude (precipitation, temperature, etc.) and the watershed’s morphology. For example, the degree of slope affects the speed of flow. Relief can be determined by means of indices or characteristic curves such as the hypsometric curve.

Hypsometric Curve

A hypsometric curve provides an overall view of the slope of a watershed, and thus the relief. This curve represents the distribution of the surface area of the watershed according to its altitude. On the x-axis, it shows the surface area (or the percentage of surface area) that lies above (or below) the altitude as represented on the y-axis. In other words, the hypsometric curve of a watershed shows the percentage of the surface area S of the watershed that is located above a given altitude H.

Figure 3.11 shows the comparison between the hypsometric curves of three water-sheds. This representation does not actually allow us to make a true comparison of the slope ratios since the watersheds are of very different sizes. One way to overcome this problem of scale, at least partially, is to standardize the altitude of the watersheds. In this case, the vertical coordinate in the hypsometric curve represents the space between the maximum and minimum altitude of each watershed. The resulting profiles make it possible to compare the reliefs of the different watersheds. (Figure 3.12)

We should add that usually, watersheds at high altitudes are characterized by preparing a hypsometric curve of the glaciers, based on measurement of the surface areas covered with ice.

Hypsometric curves are a practical tool for comparing watersheds to each other or comparing the various sections within a single watershed. They can also be used to determine the average depth of rainfall on a watershed, and provide us with some

450

Fig. 3.11 : Hypsometric curves for the Mentue catchment and two sub-catchments (Haute-Mentue and Corbassiere).

indications about the hydrological and hydraulic behavior of a watershed and its drainage system³.

The Equivalent Rectangle

The concept of the equivalent rectangle, or Gravelius rectangle, is used in calculating the slope in a watershed. This method, introduced by Roche in 1963, makes it possible to compare the slopes of different watersheds to understand the effect of slope characteristics on flow.

A rectangular watershed can be produced by the geometric transformation of the actual watershed shape, using the same surface area, the same perimeter length (or density coefficient), and therefore the same hypsometric distribution. The contour lines become straight lines parallel to the short side of the rectangle. Climate, soil distribution, the vegetal cover and the drainage density remain unchanged between the contour lines.

Let L and " represent the length and the width, respectively, of the equivalent rectangle. The perimeter of the equivalent rectangle would be equal to:

(3.6) The surface area would be equal to:

(3.7)

3.These points will be discussed further in the next chapter.

0%

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50%

60%

70%

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90%

100%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

fraction of altitude above relative altitude [%]

Mentue Haute-Mentue Corbassiere

elevation [%]

Fig. 3.12 : Hypsometric curves for the Mentue catchment and two sub-catchments (Haute-Mentue and Corbassiere).

The Gravelius shape index would be equal to:

(3.8) The three preceding equations form a system allowing us to eliminate variables P and ", giving us:

(3.9)

The solution of the system of equations induces a second order polynome that has two solutions. The second solution allows us to calculate the width of the equivalent rectangle, which is expressed as follows:

(3.10)

We will leave it to the reader to demonstrate that the product (Lx ") is equivalent to A.

The shape of the contour lines in the equivalent rectangle follows directly from the cumulative hypsometric distribution. For example, the Corbassière watershed illustrated in Figure 3.1 has a surface area of 1,952 km² and a perimeter of 8,183 km.

By applying Equation 3.5, we can determine that the Gravelius shape index of this watershed is 1.652. Next, we can use Equations 3.9 and 3.10 to determine the length L=0.54 km and the width "=3.57 km of the equivalent rectangle.

Characteristic Altitudes

The altimetric system plays an important role in describing a watershed and under-standing its hydrological behavior, because the main force affecting surface flow is gravity. In fact, several processes depend directly on altimetric characteristics; average slope, for example. The altitude of a watershed also plays a role in the control of the flow resulting from an impulse (precipitation) because altitude directly affects local climatic conditions.

Maximum and Minimum Altitudes

Maximum and minimum altitudes are obtained directly from topographic maps.

The maximum altitude represents the highest point in the watershed while minimum altitude is considered the lowest point, and is generally at the outlet. These two parameters become especially important when developing equations involving climate variables such as temperature, precipitation and snow cover. They are also used to determine the altimetric amplitude of the watershed, and are one of the parameters for calculating slope.

Average Altitude

Average altitude can be obtained directly from the hypsometric curve or from a topographic map. It is expressed as:

(3.11) where H_avg is the average altitude of the watershed [m], A_iis the area between two contour lines [km²], h is the average altitude between two contour lines [m] and A is the total area of the watershed [km²].

The average altitude is not representative of reality. However, it is sometimes used in the evaluation of certain hydro-meteorological parameters or in developing hydrological models.

The Median Altitude

The median altitude corresponds to the altitude read on the x-axis of the hypso-metric curve corresponding to 50% of the total area of the watershed (Figure 3.11).

This figure is almost equal to the average altitude when the hypsometric curve of the watershed has a regular slope: in other words, when the probability density of the watershed slopes presents a symmetrical distribution.

The Average Slope of a Watershed

Average slope is an important characteristic because it tells us something about the topography of the watershed. Slope is regarded as an independent variable. It gives a good indication of the travel time of direct runoff – and therefore of the time of concentration t_c and has a direct influence on the peak flow following a rain.

Several methods have been developed for estimating the average slope of watershed, all of them based on readings of an exact or approximate topographic map.

In the 1960s, Carlier and Leclerc proposed a method that involved calculating the weighted average of the slopes of all the surfaces located between two given altitudes.

An approximate value of the average slope can then be estimated using the following equation:

(3.12)

where i_m is the average slope [m/km] or in [^o/_oo], L is the total length of the contour lines [km] (not to be confused with the length of the equivalent rectangle), D is the equidistance between contour lines [m] and A is the surface area of the watershed [km²].

Another way of determining the average slope is using the equivalent rectangle method. In this case, the following equation can be used:

(3.13) where 'H is the maximum difference in altitude of the watershed [m], and L is the length of the equivalent rectangle [m].

As Roche (1963) noted, the calculation of average slope starting from the hypsometry of the watershed or the equivalent rectangle are just two techniques for determining a reference length that can be used to calculate average slope by using the difference in altitude. However, these calculations do not take into account the shape of the hypsometric curve, and for that reason Roche proposed the development of an index of slope i_p(see below).

It should be added, though, that it is easy to automatically calculate the average slope and determine the orientation of slopes by using digital data representing the topography of the watershed (DEM and DTM) (Section 3.5.1). We strongly encourage you to use this data.

The Slope Index

The preceding methods give good results in the case of moderate reliefs and when the contour lines are simple and evenly spaced. However, when the contour lines twist and turn, it is difficult to determine their overall length L. In order to overcome this problem and the uncertainties that result from smoothing the contour lines, Roche (1963) proposed a slope index based on the equivalent rectangle and the hypsometric curve of the watershed.

The idea is to apply the rectangle equivalent method to each contour line in the watershed, so that each contour line is transformed geometrically into parallel straight lines on the equivalent rectangle. The slope index i_p is expressed as a percentage as follows:

(3.14)

where is the fraction of total area A between two consecutive contour lines of altitude a_i-1 and a_i. L is the length of the equivalent rectangle.

The Global Slope Index

Another index also based on the equation describing the distribution of altitudes of the watershed (i.e. the hypsometric curve), is the global slope index, which is expressed in m/km and defined as:

(3.15) where H_5% and H_95% are the 5% and 95% fractiles of the hypsometric curve of the

n ~

watershed, and L is the length of the equivalent rectangle. More precisely, the altitude H_5% indicates that 5% of the area of the watershed has a higher altitude.

Usually, this index is quite close to the average slope of the watershed. For illustrative purposes, Table 3.1 summarizes the principal geometric characteristics of three Swiss watersheds (Mentue, Haute-Mentue and Corbassière).

Table 3.1 : Geometric characteristics of three Swiss watersheds (Mentue, Haute-Mentue and Corbassière).

The Index of Similarity of Hydrological Behavior

As we have seen, topography plays a determining role in the distribution of water in the soil and in the generation of flow. Using the same reasoning, we can expect that at point i in a section of soil in a watershed, the greater the surface area that drains through point i, the greater the volume of water that will pass through it. Likewise, we can expect that the less steep the slope is at point i, the smaller the motive force – essentially gravitational – of the water will be. Thus, the soil will be more likely to become saturated if the slope in the immediate area is gentler. This observation regarding the influence of topography in terms of the drained area and the local slope can be expressed with a topographic index, which is defined as the logarithmic relationship between the surface area drained per unit length of a contour line at point a_i and the tangent of the local slope at point i (Beven and Kirkby, 1979):

(3.16)

One way to represent this index is to use the complementary of the distribution of the topographic index. This equation expresses the relationship between the percentage of saturation of the watershed and the topographic index. In a strict sense, this allows us to determine the probability density of the topographic index as well as its distribution. The two equations are as follows

Characteristics Mentue Haute-Mentue Corbassière

Area [km ²] 105.0 12.5 2.0

Perimeter of watershed [km] 64.7 19.5 8.2

Length of watershed [km] 21.3 6.7 2.6

Width of watershed [km] 4.9 1.9 0.7

Gravelius index [-] 1.8 1.6 1.7

Minimum altitude [m] 445 694 848

Maximum altitude [m] 927 927 927

Average altitude [m] 679 831 885

Length of the equivalent rectangle L[km] 28.7 8.2 3.5

Width of the equivalent rectangle l[km] 3.7 1.5 0.6

Length of the main river of watershed [km] 25.7 7.3 2.8

(3.17)

(3.18)

Therefore, at time t, it is possible to determine an upper limit value of the topographic index such that all the points that have a higher topographic index are saturated. The limit value of the topographic index is expressed by the following:

(3.19) This last equation links the probability of saturation at a given point at a given time with the cumulative distribution of the topographic index of which the complement at 1 (one) of the value (calculated by equation 3.19) represents the cumulative curve of the fractions of saturated areas.