最大熵原理与地震频度-震级关系

MAXIMUM ENTROPY PRINCIPLE AND SEISMIC MAGNITUDE-FREQUENCY RELATION

地震是一种随机事件 ,它的发生具有极大的不确定性 ,因而可以用熵来进行描述。地震以最无序的方式在各地发生 ,意味着地震熵达到了极大值。古登堡 (Gutenberg)和里克特 (Richter)根据资料和经验得出的地震频度 -震级关系式实际上是在给定的约束条件下 ,当地震熵取极大值时得到的一种负指数分布。文中从最大熵原理得出了同一形式的地震频度 -震级关系 。

Entropy is a state function. Entropy increasing principle shows that under isolated or adiabatic condition the process of a system developed spontaneously from non equilibrium state to equilibrium state is a process of entropy increasing. The equilibrium state corresponds to the maximum entropy. In equilibrium state, the state of the system is most chaotic and disorder. Earthquake is a random event, the occurrence of which possesses extremely great uncertainty, and hence can be expressed by entropy. Earthquake occurs disorderly in different areas, implying that the seismic entropy has reached a maximum value. Therefore, magnitude distribution of earthquakes in one region for a certain time period can be expressed by the principle of maximum entropy. Assuming that M 0 is starting magnitude and AM-U is average magnitude, through the deduction we can get lg n= lg NAM-U-M 0+M 0AM-U-M 0-1AM-U-M 0M, where n is differential frequency, N is total number of earthquakes, and M is magnitude. The magnitude frequency relation proposed by Gutenberg and Richter according to seismic data and experience is expressed as: lg n=a-bM. Comparing the two equations gives a= lg NAM-U-M 0+M 0AM-U-M 0, b=1AM-U-M 0. Obviously, the Gutenberg Richter magnitude frequency relation is essentially a negative exponent distribution obtained by taking the maximum value of seismic entropy under a given constrained conditions. In this way the cause of magnitude frequency relation is theoretically explained.