1985 , Vol. 5 >Issue 1: 1 - 9

• 兰州大学数学系、计算站、地质地理系

A CONCENTRATED DEGREE METHOD FOR CALCULATION OF VALLEY TRENDS

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• Lanzhou University

Online published: 1985-01-20

### Abstract

Stress is physically a tensorial quantity which varies from point to point within the earth. Therefore, the stresses form a field in the earth. The present-day stress field manifests itself in a variety of phenomena, in turn, from a study and interpretation of these phenomena the stress field can be inferred and deduced. Many geomorphic features, such as orientation structure of a river network are first found by Prof. Scheidegger to be due to the action of a tectonic stress field, If the valley are supposed to be Mohr-type fractures, one obtains as predicted stress direction the bisectries of the prefferred valley trends. For calculation of the preferred directions a computational method has been developed by Kohlbeck and Scheidegger. But the program is too large for ordinary computer. The distribution pattern of valley trends is caused by the antagonistic interaction between the endogenetic and exogenetic effects. The exogentic effects give the distribution pattern randomness, meanwhile the endogenetic ones result its non randomness. According to the law of large numbers, the exogenetic effects will offset each other,the distribution will have non-random pattern. Therefore, we can research this pattern without the random statistical method, but with the method of the deterministic mathematics. In this paper we will give a new compStress is physically a tensorial quantity which varies from point to point within the earth. Therefore, the stresses form a field in the earth. The present-day stress field manifests itself in a variety of phenomena, in turn, from a study and interpretation of these phenomena the stress field can be inferred and deduced. Many geomorphic features, such as orientation structure of a river network are first found by Prof. Scheidegger to be due to the action of a tectonic stress field, If the valley are supposed to be Mohr-type fractures, one obtains as predicted stress direction the bisectries of the prefferred valley trends. For calculation of the preferred directions a computational method has been developed by Kohlbeck and Scheidegger. But the program is too large for ordinary computer. The distribution pattern of valley trends is caused by the antagonistic interaction between the endogenetic and exogenetic effects. The exogentic effects give the distribution pattern randomness, meanwhile the endogenetic ones result its non randomness. According to the law of large numbers, the exogenetic effects will offset each other,the distribution will have non-random pattern. Therefore, we can research this pattern without the random statistical method, but with the method of the deterministic mathematics. In this paper we will give a new computational method. Let θi be the direction of a river section and Li correspounding langth. A concept of concentrated degree is introduced as following: where {λ-k,......,λ0,......,λk} are a group of powers, λi-i>0, λ0 is maximum of {λi}, and if j>i, then λji. Obviously, reflects the concentrated degree of the directions of a river network at θm. The prefferred directions can be calculated by the following steps:1) compute 2) Suppose Max/n = a1, look for the maximum of , where n varies in (0°, 180°), (α1-45°, α1+45°) 3) Denote A=(α12)/2, B=A+90, then compute new concentrated degree . in and 4) Suppose β1=max , then β1 is the first preffered direction. Similarly, if β2=max, then β2 is the second preffered direction. By computing it is found that the choice of power {λi} is important. We call η=λ0k by propertion of amplitude. η can be taken near 1 when the variance of data is rather small, on the contrary, it should be taken rather more. utational method. Let θi be the direction of a river section and Li correspounding langth. A concept of concentrated degree is introduced as following: where {λ-k,......,λ0,......,λk} are a group of powers, λi-i>0, λ0 is maximum of {λi}, and if j>i, then λji. Obviously, m reflects the concentrated degree of the directions of a river network at θm. The prefferred directions can be calculated by the following steps:1) compute m 2) Suppose Max/n =al, look for the maximum of , where n varies in (0°, 180°), (α1-45°, α1+45°) 3) Denote A=(α12)/2, B=A+90, then compute new concentrated degree z. in and 4) Suppose β1=max , then β1 is the first preffered direction. Similarly, if β2=max, then β2 is the second preffered direction. By computing it is found that the choice of power {λi} is important. We call η=λ0k by propertion of amplitude. η can be taken near 1 when the variance of data is rather small, on the contrary, it should be taken rather more.

### 参考文献

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