## 1. Introduction

[2] Some properties of landscapes suggest that Earth's surface topography might be scale-invariant. Field observations and perusal of topographic maps lead to the qualitative impression that erosionally dissected landscapes have a similar appearance over a wide range of spatial scales [e.g., *Davis*, 1899; *Montgomery and Dietrich*, 1992]. Formal analyses of topographic data suggest that some landscapes may be either self-similar (consisting of landforms with the same shape and aspect ratio at every scale) or self-affine (aspect ratio varies with scale) [e.g., *Vening Meinesz*, 1951; *Mandelbrot*, 1975; *Sayles and Thomas*, 1978; *Church and Mark*, 1980; *Mandelbrot*, 1983; *Matsushita and Ouchi*, 1989; *Newman and Turcotte*, 1990; *Balmino*, 1993; *Turcotte*, 1997; *Rodríguez-Iturbe and Rinaldo*, 2001], or may display other properties of random or fractal surfaces [e.g., *Shreve*, 1966; *Ahnert*, 1984; *Culling and Datko*, 1987; *Tarboton et al.*, 1988; *Ijjasz-Vasquez et al.*, 1992; *Schorghofer and Rothman*, 2001, 2002]. These observations have led to suggestions that the physics that govern the development of erosional landforms are independent of spatial scale [e.g., *Somfai and Sander*, 1997].

[3] Yet it has also been observed that landscapes have characteristic spatial scales. Field observations and measurements show that there is a limit to the erosional dissection of landscapes, in the sense that fluvial channels begin to form at scales much coarser than the granularity of the soil [*Gilbert*, 1877, 1909; *Horton*, 1945; *Montgomery and Dietrich*, 1992; *Dietrich and Montgomery*, 1998]. Studies that report self-similarity or self-affinity of topographic surfaces often note that this property only holds within a certain range of spatial wavelengths [*Church and Mark*, 1980; *Mark and Aronson*, 1984; *Gilbert*, 1989; *Moore et al.*, 1993; *Xu et al.*, 1993; *Evans and McClean*, 1995; *Gallant*, 1997; *Dodds and Rothman*, 2000]. Many landscapes also appear to contain quasiperiodic structures, including evenly spaced rivers and drainage basins [e.g., *Shaler*, 1899; *Hanley*, 1977; *Hovius*, 1996; *Talling et al.*, 1997; *Schorghofer et al.*, 2004]. For example, the landscape in Figure 1, part of the Gabilan Mesa, California, contains NW–SE-trending, first-order valleys with a remarkably uniform spacing.

[4] The quasiperiodic valley spacing is visually striking. Is this merely a reflection of the human eye's affinity for organized patterns, or is it an important signature of the erosion processes that shaped the landscape? What is the “wavelength” of the ridges and valleys, and are they truly as nonrandom as they appear, or are they merely part of a continuum of scale-invariant landforms? Quantitative answers to these questions require a robust measurement technique that provides a statistical description of the topography.

[5] Geomorphologists often use drainage density, defined as the total length of erosional channels per unit planform area, as a measure of the extent of landscape dissection. Despite recent advances that permit the calculation of drainage density as a continuous variable across a landscape [*Tucker et al.*, 2001], drainage density cannot provide answers to all of the questions posed above. Measuring drainage density requires the mapping of channels, a task that is difficult to perform without detailed field investigations [*Montgomery and Dietrich*, 1988; *Dietrich and Dunne*, 1993], and thus drainage density cannot be reliably calculated from topographic data alone. Furthermore, drainage density measures the extent of a drainage network, but provides no specific information about its structure. A topologically random drainage network can have the same drainage density as a network with evenly spaced channels. Even in a landscape with evenly spaced first-order channels, the inverse of drainage density and the average valley spacing will be comparable [*Horton*, 1932], but not necessarily equal.

[6] If a landscape consists of well defined, parallel valleys, one can measure their spacing directly from topographic maps [e.g., *Hovius*, 1996; *Talling et al.*, 1997]. However, few landscapes have such simple structure. If valleys are not parallel, it is not obvious how or where their spacing should be measured. One therefore requires a robust measurement technique based on the overall shape of the topography rather than the planform geometry of the drainage network.

[7] Spectral analysis provides a means of measuring the strength of periodic (sinusoidal) components of a signal at different frequencies. The Fourier transform takes an input function in time or space and transforms it into a complex function in frequency that gives the amplitude and phase of the input function. If the input function has two or more independent dimensions, the Fourier spectrum gives amplitude and phase as a function of orientation as well as frequency.

[8] A number of previous studies have used Fourier transforms to analyze topographic and bathymetric data. Some of these papers discuss the identification of periodic structures [*Rayner*, 1972; *Hanley*, 1977; *Stromberg and Farr*, 1986; *Ricard et al.*, 1987; *Mulla*, 1988; *Gallant*, 1997] or textures with preferred orientations [*Steyn and Ayotte*, 1985; *Mushayandebvu and Doucouré*, 1994], whereas others use the spectrum to describe the variance structure or scaling properties of the topography [e.g., *Steyn and Ayotte*, 1985; *Voss*, 1988; *Ansoult*, 1989; *Hough*, 1989; *Goff and Tucholke*, 1997]. Many of these studies used methods that were tailored to specific data sets or questions, and thus their procedures are not readily extendible to any topographic surface.

[9] In this paper, we describe a general procedure for applying the two-dimensional, discrete Fourier transform to topographic data. We introduce a statistical method that provides a means of measuring the significance, or degree of nonrandomness, of quasiperiodic structures. By applying this procedure to two topographic data sets, we show that there are strong periodicities at certain scales, rather than a continuous distribution of spectral power across all scales, and that topographic roughness declines sharply below a certain spatial scale. We illustrate a filtering procedure that can be used to isolate the different frequency components of a topographic surface, and can thereby provide a means of measuring topographic attributes at certain scales and orientations. We conclude by discussing the implications of our results for fractal descriptions of landscapes.