شناسایی ضعف در میانگین‌گیری برداری داده‌های مغناطیدگی و روشی برای برطرف کردن این ضعف

در جوامع معمولی آماری، اغلب وضعیت توزیع عادی یا نرمال حاکم است و لذا در آنها استفاده از تابع چگالی احتمال گوسی یا نرمال و استفاده از میانگین‌گیری حسابی یا معمولی کار صحیحی می‌باشد. اما اگر جامعه آماری از تعدادی جهت دلخواه فضایی تشکیل شده باشد، وضعیت توزیع عادی یا نرمال حاکم نمی‌باشد. در این شرایط از تابع چگالی احتمال فیشر ومیانگین‌گیری برداری می‌توان بهره برد. یکی از جوامع آماری جهتی، جامعه آماری جهت های مختلف مغناطیدگی سنگ‌ها است. در این مقاله پس از یک مقدمه، برای درک بهتر تفاوت جامعه آماری معمولی و جامعه آماری جهتی، هم پراکندگی نرمال و هم پراکندگی فیشر (که برای جامعه جهتی استفاده می شود) مورد بحث قرار می گیرد. در ادامه الگوریتم محاسبه جهت میانگین مجموعه بردارها مطرح می شود. سپس به یک برنامه رایانه ای دارای توانایی میانگین‌گیری برداری که در جریان همین پژوهش تولید شده، اشاره شده است و بعد میانگین‌گیری برداری و حسابی با استفاده از داده های مغناطیدگی مقایسه شده اند. در این پژوهش معلوم شد که یک ضعف در میانگین‌گیری برداری وجود دارد و آن اینکه در بعضی شرایط جواب میانگین‌گیری برداری یکتا نمی‌باشد. راه‌حل ارائه شده در این پژوهش برای رفع این ضعف این است که در کنار میانگین‌گیری برداری، مناسب است که میانگین‌گیری معمولی یا حسابی هم صورت بگیرد تا در مواردی که جواب میانگین‌گیری برداری چند جهت متفاوت است، بتوانیم میانگین برداری صحیح را تشخیص دهیم.

Detection of the weakness of the vector averaging of magnetization data and a method for treatment of the weakness

In statistical common population, common or normal distribution is often governed and so that using Gaussian or normal probability density function and arithmetic averaging is appropriate. But if the statistical population has been formed from a number of spatial arbitrary directions, then common or normal distribution is not governed. In this condition Fisher probability density function and vector averaging can be used (Fisher is the name of the scientist who proposed the mentioned density function for the first time). In this function, each direction is shown as a point on a sphere with unit radius. The mentioned function shows the probability of having a particular direction in unit angular area of a particular area that has a definite central direction. This central direction shows the angular difference with the real average direction. In Fisher function, the distribution of the azimuth angles around the real vector average direction is symmetrical. The azimuth and the declination angles are the same and being symmetrical around the their distribution of the real average direction is logical. One of the statistical directional populations is the statistical population of different directions of the magnetization of rocks (Each magnetization direction is specified by two angles. First the angle between the magnetization direction and the surface of the horizon (inclination angle) and second the angle between the magnetization direction projection on the surface of the horizon and the geographic north direction (declination angle)). In this paper after an introduction, both normal and Fisher distributions (the latter is used for directional population) are discussed for better understanding of the difference between normal and directional statistical populations. Then the algorithm for calculating the vector averaging is presented. After that a software having vector averaging ability that is produced in this research is presented and then the vector and arithmetic averages are compared for magnetization data. During this research, it is clear that there is a weakness in the vector averaging and that weakness is that in some conditions the result of the vector averaging is not unique (this non uniqueness is because of the functions used in vector averaging algorithm). For example for calculating the declination angle, the function arctangent is used and we know that the result of this function is not unique. For example arctan (0.5637) is equal to both 29.41 and 150.9 degrees). The proposed method for the treatment of this weakness in this research is that, it would be proper to perform an arithmetic averaging beside the vector averaging and by which in the cases of having non unique results for vector averaging, the true result can be detectable (The result of the arithmetic averaging is unique) Between different results of the vector averaging, that result is true which is more similar to the arithmetic averaging. For example if there is a directional population which their declination angles are between 170 to 140 degrees and their arithmetic average is 150.67 degrees and the results of their vector averaging are 29.41 and 150.59 degrees, then the correct vector average is 150.59.