In the community of geodesy it is well-known that the famous normal distribution is originated from the mathematical analysis of observational errors in astronomical and geodetic measurements.However, as far as we know this aspect of scientific history which is of considerable interest for the community of earth sciences has rarely been considered in the literature of earth sciences.In geodesy and related areas the bivariate normal distribution is one of the most frequently used probability distributions. Nowadays, in a wide range of problems arising from diverse areas of geodesy, geophysics, photogrammetry and astronomical geodesy we encounter numerous applications of the univariate and multivariate normal distributions.In the present paper the historical role of earth sciences in the origins of the bivariate normal distribution is briefly discussed. Some new evidences of Bravais’ contribution to the origin of the correlated bivariate normal distribution are considered. The new evidences and refinements established in this paper convey such a general methodological and intellectual content that is useful for the community of geodesy, geophysics, and furthermore in earth sciences.
In recent years the uniform distributions and their convolutions find such applications that are relevant to geodesy-more precisely-to the modern theory of errors: (i) The convolutions of uniform distributions have been applied to the error distribution arising from data processing; (ii) Within the framework of geodesy, outliers were assumed to be distributed with uniform distribution. Bearing in mind these new developments and integrating these isolated topics, in this paper new closed formulae for the probability density and distribution functions of the sum of independent uniform random variables with unequal supports are derived. A brief outline of the relevance of convolutions of uniform distributions to the theory of errors related to astronomy and geodesy is given in historical setting. Along with these, the origin of uniform distribution is discussed with special emphasis on the root of the theory of errors.
Without any special term, the mathematical definition of a measuring index for reliability of geodetic point was given by Lajos Homoródi. For this index, the term of the elliptic mean error was proposed by the author of the present paper and it was shown that the elliptic mean error is beneficial to the being for characterizing the reliability of geodetic point. It is surprisingly interesting that there is a close relationship between the arithmetic-geometric mean and the elliptic mean error. This phenomenon with relevant mathematical backgrounds is presented.
We address the further improvements and clarifications to the formal and categorizing definition of outlier given by Monhor and Takemoto (2005), by means of integrating to the above definition the subclass of valuable outliers introduced by Verő (2009). The concrete illustrative examples taken from geophysics and other areas, and the further remarks on and insights into the nature of outliers presented in this paper are, to a certain extent, the contributions to the establishment of such a categorization that integrates the diverse and heterogenous appearance of outliers and helps the comprehensive grasp of the concept of outlier.