شناسايي ضعف در ميانگين‌گيري برداري داده‌هاي مغناطيدگي و روشي براي برطرف كردن اين ضعف

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

در جوامع معمولي آماري، اغلب وضعيت توزيع عادي يا نرمال حاكم است و لذا در آنها استفاده از تابع چگالي احتمال گوسي يا نرمال و استفاده از ميانگين‌گيري حسابي يا معمولي كار صحيحي مي‌باشد. اما اگر جامعه آماري از تعدادي جهت دلخواه فضايي تشكيل شده باشد، وضعيت توزيع عادي يا نرمال حاكم نمي‌باشد. در اين شرايط از تابع چگالي احتمال فيشر وميانگين‌گيري برداري مي‌توان بهره برد. يكي از جوامع آماري جهتي، جامعه آماري جهت­هاي مختلف مغناطيدگي سنگ‌ها است. در اين مقاله پس از يك مقدمه، براي درك بهتر تفاوت جامعه آماري معمولي و جامعه آماري جهتي، هم پراكندگي نرمال و هم پراكندگي فيشر (كه براي جامعه جهتي استفاده مي­شود) مورد بحث قرار مي­گيرد. در ادامه الگوريتم محاسبه جهت ميانگين مجموعه بردارها مطرح مي­شود. سپس به يك برنامه رايانه­اي داراي توانايي ميانگين‌گيري برداري كه در جريان همين پژوهش توليد شده، اشاره شده است و بعد ميانگين‌گيري برداري و حسابي با استفاده از داده­هاي مغناطيدگي مقايسه شده­اند. در اين پژوهش معلوم شد كه يك ضعف در ميانگين‌گيري برداري وجود دارد و آن اينكه در بعضي شرايط جواب ميانگين‌گيري برداري يكتا نمي‌باشد. راه‌حل ارائه شده در اين پژوهش براي رفع اين ضعف اين است كه در كنار ميانگين‌گيري برداري، مناسب است كه ميانگين‌گيري معمولي يا حسابي هم صورت بگيرد تا در مواردي كه جواب ميانگين‌گيري برداري چند جهت متفاوت است، بتوانيم ميانگين برداري صحيح را تشخيص دهيم.

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 arc-tangent 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.