geomorfГеоморфология и палеогеографияGeomorfologiya i Paleogeografiya2949-17892949-179710.15356/0435-4281-2017-2-14-24geomorf-1289Research ArticleМетодика научных исследованийScientific Research MethodsК ВОПРОСУ О  ПРИМЕНЕНИИ ГАРМОНИЧЕСКОГО АНАЛИЗА ПРИ КОЛИЧЕСТВЕННОЙ ХАРАКТЕРИСТИКЕ РЕЛЬЕФАAPPLICATION OF HARMONIC ANALYSIS FOR THE QUANTITATIVE DESCRIPTION OF EARTH SURFACE TOPOGRAPHYХарченкоС. В.KharchenkoS. V.

Kazan

Moscow

xar4enkkoff@rambler.ruКазанский (Приволжский) федеральный университет Институт географии РАН, МоскваРоссияKazan Federal University Institute of Geography RASRussian Federation20170505201702Геоморфология1424Copyright © Харченко С.В., 20172017Харченко С.В.Kharchenko S.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://geomorphology.igras.ru/jour/article/view/1289

Дан краткий обзор опыта применения гармонического анализа для морфометрической характеристики рельефа. Описаны некоторые ограничения использования аппарата дискретного преобразования Фурье (ДПФ) на топографических данных. Охарактеризована методика одномерного и двумерного преобразования Фурье топографических профилей и цифровых моделей рельефа, а именно ее математическая основа и конкретный алгоритм гармонического преобразования, реализованный в среде MathCAD. Одномерное ДПФ позволяет поставить ряду наблюдений в соответствие периодическую функцию как сумму волновых колебаний с кратными значениями частоты и периода. Для каждой такой гармоники определяются величины амплитуды и фазового сдвига. Амплитуда гармоники той или иной частоты характеризует ее вклад в колебания высот по линии профиля. Это позволяет сравнивать волны по значимости в “формировании” конкретного участка земной поверхности. Двумерное ДПФ, кроме того, позволяет выявить и направление колебаний. Идеальная фрактальная эрозионная система, в которой формы вреза разных порядков одинаково ранжируются как по ширине, так и по глубине, будет характеризоваться строгим убыванием амплитуд гармоник с увеличением их частот. Реальная земная поверхность редко проявляет такое свойство. Следовательно, преобразование Фурье может использоваться для задач классификации и районирования земной поверхности по ее гармоническим характеристикам, определяющим специфику топографического расчленения участка.

The paper is focused at perspectives of Discrete Fourier Transform (DFT) as a tool for morphometric studies. Characterized are the mathematical basics and MathCAD algorithm of one- and two-dimensional DFT when applied to topographical profiles and digital elevation models (DEMs). One-dimensional DFT allows to replace an empirical spatial series with a periodic function as a sum of waves with different frequencies and wavelengths. Amplitude of each wave describes the contribution of the corresponding frequency to the total elevation variations along the studied profile, which allows to compare different harmonics (waves of certain frequency) in terms of their importance in shaping the Earth surface. Two-dimensional DFT allows also to determine the direction of wave distribution. In an ideal fractal erosional system in which different-order thalwegs have equal ranking both by width and by depth of incision, strict decrease of harmonic amplitude with increasing spatial frequency would occur. However this feature is only rarely occurring in the real Earth surface. Consequently, the DFT method can be used for spatial classification and regionalization according to topographic patterns formally described by harmonic components of the Earth surface elevation.

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The authors declare that there are no conflicts of interest present.