Errors in linearizing the dependence of the thermal conductivity coefficient of snow on density

The subject of the research is the functional dependence of the thermal conductivity coefficient of snow on its density. The object of the research was the linearization of the function expressed by a polynomial of arbitrary degree, characteristic of the quantitative dependence of the thermal conductivity coefficient on the density of snow. Special attention is paid to the analysis of errors arising from replacing a polynomial function with a linear one. Particular attention is also given to the analysis of existing functional dependencies of the thermal conductivity coefficient on snow density, which is an integral indicator of complex heat and mass transfer processes occurring during the metamorphism of the snow cover. The main formulas for predicting the thermal conductivity coefficient based on snow density have been analyzed. All dependencies are conditionally divided into two groups: linear and nonlinear (expressed by polynomials of the second, third, and fourth degrees). To find the point corresponding to the maximum value of the linearization error of the second group of methods, an appropriate target function has been constructed and studied in its most general form. When constructing the function defining the absolute error that arises from linearization, a generalized polynomial of derivative degree, which describes known experimental and theoretical dependencies of the thermal conductivity coefficient of snow on its density, was taken as the original formulas. The resulting function was analyzed for its maximum using the classical method of differentiating the original dependence with respect to the argument. The scientific novelty lies in the fact that for the first time a dependence has been obtained between the error arising from the linear and nonlinear representations of the experimental approximating dependencies of the thermal conductivity coefficient of snow and its density. It has been shown that when linearizing a quadratic dependence (Abel's, Kondratyeva's, Brecht's, Sturm's formulas, etc.), the maximum absolute error is found in the middle of the averaging interval. At this point, its value equals the value of the original function. As the degree of the polynomial increases, the maximum error shifts toward the upper boundary of the linearization interval and changes, for example, for the cubic polynomial (Van Dussen's formula) to a value equal to 0.58 of the linearization range. For the fourth-degree polynomial (Jansen's formula), it reaches 0.63 of the range. When the degree is reduced to less than two (Yen's, Schwander's formula), on the contrary, the maximum linearization error shifts from the middle of the interval to the lower boundary.

snow, properties, density, thermal conductivity, coefficient, formula, linearization, forecast, error, accuracy of determination