Linear
Regression in HydrologyLinear
Regression in HydrologyStreams are a dynamic phenomena and one of the best methods to understanding streams is the use of statistics. Often the amount of energy a stream has at any point is to be determined. The energy a stream has is mostly lost as heat. A small fraction of the total energy is used to erode, deposit and transport clay and silt. Schumm(1) has determined a relationship between width/depth ratio and clay/silt content. Since the width/depth ratio is so closely linked to the stream gradient, it is postulated in this paper that there is a direct relationship between clay/silt content and stream gradient.
As energy is used in different amounts to erode, deposit and transport clay than to process silt a measure of the clay in proportion to silt is needed. Schumm(2) has done this with a scale indicating precentage content. These processes in turn will determine the amount of cutting and therefore the stream gradient.
To record the amount of silt and clay sieves are used. Sampling would be along the entire lenght of the stream, from headland tributarys to mouth. At each point that measurements are taken a measure of the stream gradient would also be obtained. This could be done with surveying techniques as the gradient is an angle with the horizontal. The result would be two sets of data each on the interval scale. While the data need only be taken from one stream the value would be universal. Meaning all streams would display this relationship. At least twenty points along the stream would be needed.
These measured values seperately should have normal distributions. Kurtoisis of each set would need to be close to three. Also the skew with the clay/silt content values would probably need to be logged, because of the range. The gradient values follow patterns of the terrain and river bed. This relationship and the distance between measuring points determines the distribution. In other words data collection insures that the data is normalized.
The graph would show clay/silt content from 0 - 100% along the bottom. The gradient might measure from 0 -90 degrees but to be realistic one would expect a much smaller scale, more like 0 - 10 degrees along the Y axis. Using an elipse the strength of the relationship could be determined. A Spearman rank order test could lead to more information about the exact relationship. Finally a linear regression table and line could be constructed. Along with this line one would determine the mean centre.
The standard error of the regression line should also be plotted. Using three values of X (the gradient) the limits of the true regression line can be found. Also for a few well placed values of clay/silt content the standard error of estimates could be determined using:
Standard error of estimates =
Since this determined relationship will be used to predict values of the stream gradient the last values to be calculated will be the 95% confidence limits of indivdual values of Y.
After this work has been completed the exact relationship between stream gradient and clay/silt content will be known. The previous work done with clay/silt content and stream width/depth ratio also involved a linear regression(3). Many observable relationships between stream properties are direct linear relationships. Sometimes the scales must be converted to logarithms. The reason a linear regression would be used is that it is expected that these variables will relate in a linear fashion. There is a chance that the relationship is curvilinear or of another type. Curvilinear relationships require another type of statistical test and would probably be incountered in the study of streams.
Footnotes:
1. M. Morisawa, Streams: thier dynamics and thier morphology, Mgraw-Hill.
2. ibid.
3. ibid.
Copyright Peter Timusk
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