4.1. Deviations From Fractal Scaling
 Having demonstrated the application of Fourier analysis to high-resolution topographic data and explored the information that can be extracted, we can return to the question of whether Earth has fractal surface topography. As noted in section 1, previous authors using similar techniques have drawn the conclusion that topography is scale-invariant. In this section, we evaluate this conclusion by comparing our results with the predictions of the fractal model.
 Landforms are generally larger in amplitude at longer wavelengths. The simplest spectrum with this property is a “red noise” spectrum with an inverse power-law dependence of spectral power on frequency: P(f) ∝ f−β. It is commonly reported that topographic spectra obey this relationship, and considerable attention has been devoted to interpretations of the exponent β [e.g., Burrough, 1981; Mark and Aronson, 1984; Hough, 1989; Norton and Sorenson, 1989; Huang and Turcotte, 1990; Polidori et al., 1991; Chase, 1992; Klinkenberg and Goodchild, 1992; Lifton and Chase, 1992; Ouchi and Matsushita, 1992; Xu et al., 1993; Gallant et al., 1994; Wilson and Dominic, 1998]. In general, β reflects the rate at which the amplitudes of landforms decline relative to wavelength. For two-dimensional spectra, β = 3 indicates that amplitude is directly proportional to wavelength [Voss, 1988], such that landforms are self-similar, with a height-to-width ratio that is independent of scale. Other values of β imply that the topography is self-affine rather than self-similar: β > 3 indicates that shorter-wavelength features have smaller height-to-width ratios, and β < 3 indicates that shorter-wavelength features have larger height-to-width ratios. The exponent β is related to the fractal dimension, D, of the surface by [Berry and Lewis, 1980; Saupe, 1988; Huang and Turcotte, 1990]
Note that these relationships apply to the spectra in Figures 4a and 4b because they are collapsed versions of two-dimensional spectra, as opposed to spectra derived from one-dimensional topographic profiles.
 A topographic surface that is well described by the fractal model has some notable properties. First, the same scaling relationship between amplitude and wavelength should hold over all wavelengths. Second, the fractal dimension of a surface should lie within the range 2 ≤ D ≤ 3. From equation (14), the exponent β in the relationship P(f) ∝ f−β should therefore lie within the range 2 ≤ β ≤ 4. Third, there should be no concentration of variance into particular frequency bands, and therefore the topography should consist of landforms with a continuum of wavelengths.
 The Gabilan Mesa and Eel River display several spectral characteristics that are inconsistent with the fractal model. First, the kink in the power spectrum, with a rapid decline of spectral power at higher frequencies (Figures 4a and 4b) implies a transition to a different scaling relationship between amplitude and wavelength at wavelengths less than ∼180 m for the Gabilan Mesa, and ∼250 m for the Eel River. At intermediate frequencies below this spectral roll-off, the spectral slopes for the Gabilan Mesa (β = 2.8) and the Eel River (β = 3.1) are close to 3, indicating that landforms have a nearly constant height-to-width ratio. Above the roll-off, the steeper spectral slopes (β = 5.2 and 4.5, respectively) indicate a height-to-width ratio that declines with increasing frequency. This does not imply that there are no topographic features at scales below the spectral roll-off, nor does it necessarily imply that dissection of the landscape by channel networks does not proceed at finer scales. It does imply that finer-scale features are much smoother than coarser-scale features. The observation that the break in spectral slope is larger for the Gabilan Mesa than for the Eel River suggests that short-wavelength features make a somewhat larger contribution to the topographic roughness at the Eel River, and that the transition from a landscape composed of ridges and valleys to one composed of relatively smooth hillslopes is more pronounced in the Gabilan Mesa.
 Several previous studies have noted a similar decline in spectral power at short wavelengths, and although some of these conclude that it is probably an accurate reflection of the shape of the topography [Culling and Datko, 1987; Gallant, 1997; Gallant and Hutchinson, 1997; Martin and Church, 2004], it is often interpreted as an artifact of topographic data interpolation [Polidori et al., 1991; Moore et al., 1993; Gallant et al., 1994]. This clearly is not the case at the Gabilan Mesa or the Eel River sites, because the 4-m topographic data can resolve frequencies much higher than the roll-off frequency. Spectral evidence for a lower limit of topographic roughness may have been overlooked in the past because of the low spatial resolution of topographic data. The kink in the spectrum may not be as apparent in previously published spectra because it occurs at a frequency comparable to the Nyquist frequency of many topographic data sets. For a 30-m digital elevation map, for example, the shortest resolvable wavelength (equal to the inverse of the Nyquist frequency) is 60 m, which is only a factor of 2 to 3 smaller than the wavelengths at which the spectral transitions in Figures 4a and 4b occur. In contrast, the high-resolution data used here leave little doubt that the spectral kink represents a change in the character of the topography. This observation underscores the need for topographic data with a resolution sufficient to reveal landscape structure at scales significantly finer than that of first-order drainage basins.
 It is sometimes suggested that a break in the scaling properties of a topographic surface indicates a transition from one suite of scale-invariant physical processes to another, with a resultant transition in the fractal dimension of the topography [e.g., Huang and Turcotte, 1990]. If we attempt to apply this concept to the topographic spectra presented here, we find a second way in which they are incompatible with the fractal model. As mentioned above, the exponent β for a fractal surface should lie between 2 and 4. Both the Gabilan Mesa (β = 2.8) and the Eel River (β = 3.1) satisfy this constraint at frequencies below the roll-off (though the range of frequencies below the roll-off is too narrow to give a clear fractal spectrum), but at frequencies above the roll-off, both spectra exhibit power-law scaling trends with β > 4. This demonstrates that both landscapes are smoother at fine scales than a fractal surface.
 The characteristic of the two landscapes that is most at odds with the fractal model is the occurrence of quasiperiodic ridge-and-valley structures in the topography. The resulting concentration of power into specific frequency bands can appear small when spectra are plotted on logarithmic axes, particularly when spectral power spans many orders of magnitude, but the significance of the spectral peaks becomes more apparent when compared with an appropriate background spectrum (Figures 4e and 4f). Indeed, we have shown that much more variance is concentrated into these frequency bands than would be expected for a random surface.
 Figures 4e and 4f also show a well-defined break between the spectral peaks associated with quasiperiodic structures at different wavelengths. The spectral decomposition illustrated in Figure 5 demonstrates that these peaks in the spectrum correspond to the roughly orthogonal ridges and valleys observed in the shaded relief maps of the topography. This is consistent with the visual impression that there is a break in scale between successive branches of the valley network in both landscapes.
 The results presented here, which are based on only two study areas, do not necessarily indicate that the fractal model is inconsistent with all landscapes. Many landscapes have less uniform spacing of ridges and valleys than the Gabilan Mesa. The presence of complicating factors such as local tectonic deformation, heterogeneities in the strength or structure of bedrock and soil, or complex boundary conditions can obscure the characteristic scales that would otherwise emerge in an evolving landscape. For example, in the northwestern portion of the landscape in Figure 1, where several roads wind along the hillslopes, localized tectonic deformation has resulted in a pattern of landscape dissection that appears less periodic than the remainder of the terrain. In some cases, mechanisms such as bedrock jointing may introduce characteristic scales that differ from those expected from erosion processes alone.
 Characteristic scales can emerge from erosion processes even in landscapes where significant heterogeneities exist, however. The steep terrain surrounding the South Fork Eel River is sculpted by debris flow scour, slope-dependent hillslope sediment transport, and deep-seated landsliding. The landslides disrupt the ridge-and-valley topography in many areas, leaving irregular topographic benches and discontinuous valleys. These effects are apparent in the eastern and northeastern portions of the area shown in Figure 3b. As we have shown here, characteristic scales still emerge in some parts of the landscape, but the prominence of deep-seated landsliding at the Eel River is probably one of the reasons why the topography there is less periodic than that of the Gabilan Mesa (section 3.1.3).
 While the occurrence of characteristic scales in landscapes is by no means universal, our measurements imply that they might be present in landscapes that were previously thought to be scale-invariant. Field observations suggest that quasiperiodic structures are common in landscapes in which erosion processes, substrate properties and tectonic forcing are spatially uniform. In the Gabilan Mesa, for instance, the topography has been produced by the dissection of poorly consolidated Plio-Pleistocene sediments with bedding planes parallel to the original mesa surface. These sediments and the granitic basement beneath them have been uplifted with minimal local deformation, and the base level for the Mesa is set by the incision of the Salinas River to the southwest [Dohrenwend, 1975; Dibblee, 1979]. Models of long-term landscape evolution, which explore the interactions of erosion processes with simple tectonic forcing, geometrically simple boundary conditions, and spatially uniform substrate properties, support the idea that quasiperiodic landforms can develop under such conditions [e.g., Howard, 1994; Kooi and Beaumont, 1996; Densmore et al., 1998; Tucker and Bras, 1998].
 Such self-organized features inspired some of the earliest hypotheses about landscape evolution mechanisms. Davis  and Gilbert  suggested that the transition from hillslopes to valleys is controlled by a transition in process dominance from slope-dependent transport (creep) at small scales to overland flow transport at larger scales, an idea that was expounded on quantitatively by Kirkby . Smith and Bretherton  and others extended this idea of a process competition to the incipient development of spatially periodic landforms, but these studies did not make predictions that could be compared to field measurements. Horton  introduced the idea that a threshold for overland flow erosion sets the scale of the hillslope-valley transition by creating a “belt of no erosion” on and around drainage divides.
 In a separate manuscript (J. T. Perron et al., Controls on the spacing of first-order valleys, submitted to Journal of Geophysical Research, 2008), we build on these previous analyses to investigate the origins of the characteristic scales documented here. Using a dimensional analysis approach combined with a numerical landscape evolution model, we demonstrate that the wavelength of quasiperiodic ridges and valleys depends on the spatial scale at which fluvial dissection gives way to smooth hillslopes, and on the relative rates of the dominant erosion and transport processes shaping soil-mantled landscapes like the Gabilan Mesa. Our analysis indicates that it is possible to derive quantitative estimates of long-term process rates by measuring characteristic scales of landscape self-organization.
4.2. Benefits and Limitations of the Fourier Transform
 The examples we have presented demonstrate that spectral analysis is a robust means of analyzing topographic structures that are qualitatively apparent but difficult to measure objectively. One could use a map and ruler to measure the spacing of some of the subparallel valleys in the Gabilan Mesa, but such an approach involves an arbitrary choice of which valleys to measure, and is poorly suited to landscapes in which the ridges and valleys are not parallel. In contrast, spectral analysis provides the basis for a relatively simple, accessible measurement technique that (1) reflects the entirety of a sample of terrain rather than a few features selected because they are visually striking, (2) is sensitive to elevation in addition to the horizontal structure of the topography, (3) can be applied to landscapes with variable ridge and valley orientations, and (4) requires no subjective delineation of landscape elements, such as the extent of the channel network.
 There are two main problems with the application of the discrete Fourier transform to topographic data. First, the data are usually nonstationary, even when periodicities are as pronounced as in the Gabilan Mesa. Indeed, nonstationarity of the signal may be one attribute of topography that contributes to apparent fractal scaling [Hough, 1989]. Second, topographic features such as ridges and valleys are not sinusoids, but instead have a complex shape that must be described by a range of frequencies.
 Techniques have been developed to address these problems. The maximum entropy method [Burg, 1967, 1975; Press et al., 1992, section 13.7] is sometimes used to estimate the power spectrum of nonstationary data sets of short duration or small spatial extent. Wavelet transforms allow for a variety of nonsinusoidal basis functions, and were designed with nonstationary signals in mind. They have been applied in a variety of fields in which nonstationary signals are common (for reviews, see Foufoula-Georgiou and Kumar  and Kumar and Foufoula-Georgiou ), including topographic analysis [e.g., Malamud and Turcotte, 2001; Lashermes et al., 2007]. A branch of wavelet analysis using basis functions better suited to topographic surfaces has been applied to one-dimensional topographic profiles [Gallant, 1997; Gallant and Hutchinson, 1997], and wavelets have proved useful for identifying morphologic transitions similar to those documented here [Lashermes et al., 2007].
 These techniques have limitations, however. The maximum entropy method is subject to the same effects of nonstationarity as DFTs, and so the lone advantage of the technique in this context is that it allows nonstationary data sets to be parsed into shorter segments for analysis. The results of wavelet transforms (particularly transforms of two-dimensional data) are more difficult to interpret than those of the Fourier transform, and the positive wavelet transforms that use basis functions modeled after landforms are nonreversible [Gallant, 1997; Gallant and Hutchinson, 1997], making filtering impossible. Our results demonstrate that by using the preprocessing steps described here, it is possible to make useful measurements with the discrete Fourier transform, which is relatively simple to apply, produces results that are easily interpreted, and can easily be extended to filtering applications. Use of the Fourier transform also facilitates comparisons with past research on the scaling properties of landscapes, many of which have been based on Fourier spectra.
4.3. Further Applications in Geomorphology
 Spectral analysis of topography has several applications beyond those presented above. By performing DFTs within a moving window, it would be possible to map the spatial variability in landscape properties, such as the wavelength or significance level of quasiperiodic structures, in a manner analogous to that used to produce the continuous map of local relief in Figure 6. Spectral properties could provide a basis for comparing attributes of synthetic topography with those of natural landscapes. For instance, temporal variations in the power spectra of numerical or physical models of landscape evolution could be used to quantify the approach to a statistical steady state when an exact steady state (fixed topography in which the erosion rate is spatially constant) is not observed. Laboratory experiments [e.g., Hasbargen and Paola, 2000; Lague et al., 2003] have produced topographic surfaces that reach a mass-balance steady state, but in which elevation is not a constant function of position and time. Because the power spectrum contains no phase information, it should remain unchanged if the frequency content of the model landscape is the same, even if the positions of ridges and valleys are not fixed. Finally, the observation that much of the variance in high-resolution topographic data is concentrated in relatively narrow frequency bands highlights the potential for data compression techniques that store and transfer topographic information as a function in the frequency domain rather than in space, an approach analogous to widely used compression standards for digital images [e.g., Wallace, 1991].