AVERAGING IN MULTIFREQUENCY SYSTEMS WITH DELAY AND LOCAL INTEGRAL CONDITIONS

Multifrequency systems of dierential equations were studied with the help of averaging method in the works by R.I. Arnold, Ye.O. Grebenikov, Yu.O. Mitropolsky, A.M. Samoilenko and many other scientists. The complexity of the study of such systems is their inherent resonant phenomena, which consist in the rational complete or almost complete commensurability of frequencies. As a result, the solution of the system of equations averaged over fast variables in the general case may deviate from the solution of the exact problem by the quantity O (1). The approach to the study of such systems, which was based on the estimation of the corresponding oscillating integrals, was proposed by A.M. Samoilenko, which allowed to obtain in the works by A.M. Samoilenko and R.I. Petryshyn a number of important results for multifrequency systems with initial , boundary and integral conditions. For multifrequency systems with an argument delay, the averaging method is substantiated in the works by Ya.Y. Bihun, R.I. Petryshyn, I.V. Krasnokutska and other authors. In this paper, the averaging method is used to study the solvability of a multifrequency system with an arbitrary nite number of linearly transformed arguments in slow and fast variables and integral conditions for slow and fast variables on parts of the interval [0, L] of the system of equations. An unimproved estimate of the error of the averaging method under the superimposed conditions is obtained, which clearly depends on the small parameter and the number of linearly transformed arguments in fast variables.

Stratigraphic mapping of sedimentary formations in southern Ontario by ground electromagnetic methods

Multifrequency and transient electromagnetic (EM) soundings were done at several locations in southern Ontario to evaluate the possibility of stratigraphic mapping of gently dipping sedimentary formations by ground EM techniques. The possibility of detecting structural features such as folds, faults, and grabens in the formations was also explored. The two EM techniques were used to map a buried river valley near the village of Copetown, about 75 km southwest of Toronto. The valley, buried under fairly resistive glacial till and dolomite formations was mapped at depths ranging from 100 to 200 m. The interpreted shape and depth of the valley agreed well with information from a high resolution seismic reflection survey and data from wells penetrating the bedrock. Transient EM (TEM) data, using square transmitter loops with side dimensions comparable to the transmitter‐receiver separation for the multifrequency system, was found to be less affected by the presence of shallow conductors and lateral inhomogeneities than the multifrequency data, and had greater depth of investigation. TEM soundings were made on five additional profiles in the area. Lithologic logs from old oil and gas wells located near the profiles provided information for comparison with EM sounding interpretations. The depths to various dolomite, shale, and limestone formations interpreted from EM data agreed well with drillhole information. The EM soundings also detected structures in the Paleozoic formations such as faults and folds that were not previously known. The soundings confirmed the existence of a graben at one site at a depth of 100 m, which was postulated from logs from a cluster of closely spaced drillholes by geologists. The survey results indicated that EM sounding methods can be used for stratigraphic mapping in areas where detailed geological information is unavailable either because the bedrock is concealed by overburden, or when drillholes are sparsely distributed.