~lathematical Geology, Vol. 9, No. 2, 1977

Fourier Analysis of the Catawba Mountain Knolls, Roanoke County, Virginia 1

J. Thomas Hanley 2

I N T R O D U C T I O N

The Catawba Mountain knolls are northwest of Roanoke in southwestern Virginia, on the Glenvar 7�89 minute quadrangle. The intermittent streams that bound each of these knolls show a striking regularity of spacing (Fig. 1). F rom a study of the interknoll spacings, it was found by means o f a chi-square test that the distribution of the spacing was not significantly different from a normal distribution having a mean of approximately 240 m and standard deviation of approximately 60 m (Hanley, 1976). A topographic profile showing this degree of regularity has not, to the author ' s knowledge, been analyzed before for the underlying harmonics by use of the Fourier transform.

The data were generated by drawing a line across the knolls and taking elevations at a regular spacing of 36 m. A total of 345 measurements were taken.

ANALYSIS OF DATA

The topographic profile of the Catawba Mountain knolls was analyzed by means of the one-dimensional Fourier transform. When the transform was applied, a set of coefficients was generated which describes the amplitude and phase of the complex frequency function that represents the topographic profile. The values of the coefficients can be used to assess the contribution of the various harmonic components. The number of coefficients generally required depends on both the sample spacing and the length of the profile.

The mathematical calculations were performed using the F I L T R A N computer program documented in Cohn (1975). F I L T R A N calculates the one-dimensional Fourier t ransform which is valid for time or distance functions of finite length.

x Manuscript received 25 June 1976; revised 31 August 1976. 2 United States Geological Survey, Reston, Virginia 22092.

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Fourier Analysis of the Catawba Mountain Knolls 161

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. . . . . ~ . . . . . . . . . . . . / , / c , _ . _ _ u _ o , L

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Figure 2. Absolute amplitude spectrum of Catawba Mountain knolls profile showing cutoff.

Filtering was used to subdue the amplitude of certain frequencies in the input data. The filtering was performed in the frequency domain. This involved multiplying the transform frequencies by 0 or 1, depending on whether a given frequency was to be removed or retained.

The relative amplitude spectrum (Fig. 2) shows the relative contribution of each frequency to the total variation of elevation in the topographic profile. The zero frequency or average elevation has been eliminated from the spectrum. The amplitude associated with 8 sampling intervals which equals 288 m was used as the cutoff for subsequent filtering as this represents the high-amplitude peak that has a wavelength closest to 240 m, the mean of the interknoll distance. This distance is within 1 standard deviation (60 m) of the mean of the interknoll distances. Any frequency whose amplitude fell below the cutoff value was removed. To preserve the continuity of the Fourier series, the peaks above the cutoff must retain 1 or 2 wavelengths on either side after filtering. Each collection of wavelengths is called a wavelength packet. The large peak at the extreme left of the spectrum was filtered out because it represents the transform of the ends of the profile and long wave- lengths that mask the shorter wavelengths.

The wavelengths below the cutoff and not included in the wavelength packets were removed. The resulting reconstructed profile is shown in Figure 3B. There were 27 of the original 345 wavelength components included in this reconstruction. The reconstructed profile bears a close resemblance to the original profile shown in Figure 3A.

Essentially 4 wavelength packets are in the absolute amplitude spectrum. Each of these, isolated by removing the other wavelengths in the spectrum, were used to produce partial reconstruction of the profile. Profiles for pairs of wavelength packet profiles are shown in Figure 3C and Figure 3D. These wavelength packets added together yield the reconstructed distance function.

D

162 J. Thomas Hanley

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Figure 3. Catawba Mountain knolls topographic profile and reconstructed Fourier series after filtering. A: topographic profile; B: reconstructed Fourier series; C: 432-m and 576-m wavelength packets; D: 180-m and 288-m wavelength packets. Relative heights correspond to minimum elevation (= 0) and maximum elevation (= lOO).

The 4 wavelength packets correspond to multiples of 5, 8, 12, and 16 ol the sampling interval (36 m), which equal 180, 288,432, and 576 m respectively.

It is interesting to note that the average of the 180- and 288-m cycles is 234 m, which is close to 240 m, the mean of the interknoll distances. Also, the average of the 432- and 576-m cycles is 504 m, which is approximately twice 240 m.

The two short wavelength cycles (180 and 288 m) obtained from the one- dimensional Fourier analysis are within one standard deviation of the mean of the interknoll distances. The two larger wavelength cycles (432 and 576 m) are inconsistent with the mean of the measured distances; however, it might be possible to reduce this inconsistency by applying a different filter or by changing the sampling interval.

SUMMARY

This note is an example of the application of Fourier analysis to the topo- graphic representation of land forms. The technique provides a quantitative

Fourier Analysis of the Catawba Mountain Knolls 163

description of topographic profiles. It alleviates the necessity of defining profiles by a series of points and allows profiles to be modeled with greater ease. Surfaces could be analyzed by two-dimensional Fourier analysis. Thus Fourier analysis is a potentially useful technique for comparing the topo- graphy of different areas quantitatively.

REFERENCES

Cohn, B. P., 1975, A forecast model for Great Lakes water levels : unpubl, doctoral disserta- tion, Syracuse Univ., 235 p.

Hanley, J. T., 1976, Structural geomorphology of the Catawba Mountain knolls, Virginia: unpubl, masters thesis, Syracuse Univ., 80 p.