Quaternary alluvial fans have diverse surface morphologies related to both depositional and weathering processes. Numerous studies have demonstrated that the surface expression and morphometry of alluvial fans can be used as an indicator of their relative age of deposition, but only recently has there been an effort to utilize high-resolution topographic data to differentiate alluvial fans by surface age with automated and quantifiable routines. We developed a quantitative model for mapping the relative age of alluvial fan surfaces based on multiparameter surface roughness values computed from 1 m resolution lidar topographic data. Roughness is defined as a function of observational scale and integration of slope, curvature, and aspect topographic parameters. Alluvial fan roughness values were computed across multiple observation scales (3 × 3 m to 150 × 150 m) based on the standard deviation (SD) of slope, curvature (tangential), and aspect topographic parameters. Plots of roughness value versus size of observation scale suggest that the SD of each parameter over a 7 × 7 m observation window best identified the signature of surface roughness elements. Roughness maps derived from slope, curvature, and aspect at this scale were integrated using fuzzy logic. The integrated roughness map was then classified into five relative morphostratigraphic surface age categories (active wash to ~400 ka) and statistically compared with a similar fivefold surface age map of alluvial fans developed using traditional field surveys and aerial photographic interpretation. The model correctly predicted the distribution and relative surface age of ~61% of alluvial fan landforms based on traditional mapping techniques.
 Alluvial fans have long been recognized as an important record of Quaternary (1.8 Myr to present) climate and tectonic activity across arid to semiarid deserts of North America [Gilbert, 1877; Davis, 1905; Blackwelder, 1931; Denny, 1965; Bull, 1977, 1984, 1991, 2008; Wallace, 1977; Wells et al., 1987; Lubetkin and Clark, 1988; Whipple and Dunne, 1992; Ritter et al., 1993; Bierman et al., 1995; McDonald et al., 2003; Matmon et al., 2005; Nichols et al., 2006; Bacon et al., 2010a]. Mapping the spatial distribution of alluvial fans of different ages is a critical part of deciphering Quaternary geologic history, as well as elucidating key ecosystem processes in arid environments [Sweeney et al., 2011]. A key quality of alluvial surfaces is that they can be stratigraphically subdivided by their local relief or height above the active channel, degree of dissection, drainage pattern, soil characteristics, and development of desert pavement [Christenson and Purcell, 1985; McFadden et al., 1989; Bull, 1991; McDonald et al., 2003; Bacon et al., 2010a].
 Many qualitative and quantitative techniques have been used to differentiate or subdivide sequences of alluvial fan surfaces. The most common techniques include (1) mapping characteristics of a fan surface in the field by describing surface clast size, rock varnish accumulation, desert pavement development, stratigraphic relationships, and evaluation of surface morphology [Colman and Pierce, 1986; Wells et al., 1987; McFadden et al., 1989; Bull, 1991; Ritter et al., 1993; Birkeland, 1999]; (2) using soil stratigraphy and the relative degree of soil development [McFadden et al., 1989; Bull, 1991; McDonald et al., 2003; Bacon et al., 2010a]; (3) mapping fan surfaces based on the difference of the surface brightness manifested in aerial or satellite images [Christenson and Purcell, 1985; Bull, 1991]; (4) analysis of remotely sensed multispectral images [Alwash et al., 1986; White, 1993; Farr and Chadwick, 1996; Beratan and Anderson, 1998], multichannel thermal infrared images [Gillespie et al., 1984], hyperspectral images [Crouvi et al., 2006], and radar images [Kierein-Young, 1997]; (5) analysis of digital elevation models [Frankel and Dolan, 2007]; and (6) application of cosmogenic age dating techniques to determine the ages of the alluvial deposits and the rates of arid-region alluvial processes [Nichols et al., 2002, 2006; Matmon et al., 2006].
 Each of the above approaches has important benefits and limitations. Traditional field mapping techniques provide the highest level of accuracy; however, field-based techniques can be labor intensive, time consuming, and difficult for mapping regional areas. Mapping based mostly on aerial or spaceborne imagery can be subjective with the accuracy of the map depending on the expert's prior knowledge, map scale, and the quality and resolution of images that show contrasts between different alluvial fan surfaces. Remote sensing techniques also have limitations. The spectral and spatial resolutions of multispectral sensors are not sufficient to explain the surface variability of alluvial deposits. The range of reflectance of alluvium is influenced by soil geomorphic surface processes including varnish and pavement development, degree of dissection and weathering, and lithology of source materials [McDonald, 1994]. Alluvial deposits of similar age may contain sediment of different compositions, which can influence rates of varnish development and weathering, thereby affecting the accuracy of interpreting remotely sensed data.
 One technique being increasingly applied to differentiate alluvial fan surfaces is to quantify the expression of alluvial surface roughness using digital topographic data [e.g., Kierein-Young, 1997; Frankel and Dolan, 2007]. Use of digital topographic data to identify fan characteristics is time efficient and can be cost effective when applied over a large area; however, its relative accuracy compared to traditional mapping methods remains uncertain. Surface morphology is a time- and process-dependent feature widely used to distinguish alluvial fan types because (1) fan surfaces initially tend to become smoother with increasing age due to the formation of desert pavement and the degradation of bar-and-swale topography and (2) subsequently, landforms become more dissected due to tectonics and climate change induced increased erosion and channelization of the fan surface with time [e.g., Wells et al., 1987; Bull, 1991; Ritter et al., 1993; McDonald, 1994; Frankel and Dolan, 2007].
 The overall focus of this paper is twofold. First, we evaluate the potential application of multiparameter surface roughness values to automatically map alluvial fan stratigraphy using high-resolution lidar (light detection and ranging) topographic data. Second, we compare model-based maps computed from multiparameter surface roughness values with alluvial fan stratigraphy mapped using traditional field and image analysis techniques. We present a way to quantify roughness by analyzing three geometries (slope, curvature, and aspect) of surface irregularities within different scales of observation. This approach differs from other studies [e.g., Frankel and Dolan, 2007] where only a single geometric measure, slope, was used to characterize the surface roughness of alluvial fans. A number of other approaches to quantifying landscape roughness [e.g., McKean and Roering, 2004; Booth et al., 2009; Hurst et al., 2013] also exist in the literature; however, these approaches have yet to be tested on alluvial fan landforms. The specific goal of this study is to assess the ability of lidar elevation data (1 m horizontal resolution) to separate different aged alluvial fan surfaces in a manner comparable to the subdivision of alluvial fans based on traditional mapping techniques. To achieve this goal, the following objectives were met: (1) prepare a geomorphic map of alluvial fan stratigraphy based on traditional field data and image analysis, (2) define and quantify the degree of roughness of alluvial fans at multiple scales using lidar elevation data, (3) determine a scale most appropriate to characterize the roughness of alluvial fan surfaces, and (4) directly compare the alluvial fan stratigraphy derived from traditional techniques with the alluvial fan stratigraphy derived from modeled surface roughness in order to evaluate the relative accuracy of a multiparameter roughness approach.
2 The Study Area
 The study area is located in the Sonoran Desert of southwestern Arizona approximately 40 km northeast of Yuma, Arizona (Figure 1). The studied area (hereafter simply “Yuma”) covers ~60 km2 area of extensive alluvial fans that are just north of the Muggins Mountains. The area has an arid climate with a mean annual precipitation of 93 mm based on precipitation data recorded between 1958 and 2012 [Western Regional Climate Center, 2012]. The study area is dominated by low-gradient and broad alluvial fans that cover most of the lowland area between intervening mountain highlands. The principal sources of fan deposits in and around the study area are moderate to low relief mountains composed mostly of Cretaceous and Tertiary granitic and volcanic rocks and lesser amounts of sedimentary rocks [Richard et al., 2000].
2.1 Surface Morphology of Alluvial Fans in Yuma
 The ages of alluvial fans vary greatly, ranging from active wash (Map unit: Qf5) to ~400 ka (Qf1) (Table 1 and Figure 2). The younger Holocene alluvial fan surfaces have well-developed bar-and-swale microtopography with variable relief (1–2 m) that is predominantly controlled by extremely gravelly soils with a range of mixtures of boulders to cobbles and pebbles. The microtopography lacks well-developed desert pavement. In contrast, Pleistocene alluvial fan surfaces exhibit smooth surface morphologies and well-developed varnished desert pavements. The oldest Pleistocene fan surfaces are incised by a network of channels that grade to Holocene age alluvial surfaces [Lashlee et al., 2002; Nichols et al., 2006; Bacon et al., 2010a]. The fans in Yuma have a range of surface morphologies at local or microtopographic scales (<10 m planimetric observation length) that are related to (1) development of desert pavement, (2) occurrence of bar-and-swale microtopography, (3) vegetation density and associated plant mounds and plant scars [McAuliffe and McDonald, 2006], and (4) the pattern and density of active channels and the sizes of their bed loads. Similar to other arid environments, the older fan surfaces in the study area are largely smoother than the surfaces of younger fans (Figure 3). These older (Pleistocene) fan surfaces are mostly devoid of vegetation and lack bar-and-swale microtopography. The younger (Holocene) alluvial fan surfaces consist of relatively greater vegetation densities in and around active channels and washes.
Table 1. Summary of Geomorphic Characteristics of Alluvial Fans Within Yuma, Arizona
Geologic age from Lashlee et al. , McAuliffe and McDonald , Nichols et al. , and Bacon et al. [2010a].
Active wash and floodplain
Late Holocene to active (1 to 0 ka)
Distributary, anastomosing. or braided with well-developed bar-and-swale microtopography
Very low (<1 m)
High (on surface and channel margins)
Surface dominantly composed of boulders, pebbles, and cobbles
Late Holocene (3.2 to 2.9 ka)
Distributary, anastomosing, or braided with well-developed bar-and-swale microtopography
Low (1–2 m)
Weak to moderate (where clasts reworked from older surfaces)
High to moderate (on surface and channel margins)
The pattern of channels is similar to Qf5 unit but lacks the relief
Late Pleistocene to early Holocene (15 to 8 ka)
Distributary to dendritic with moderately to variably developed bar-and-swale microtopography
Moderate (2–3 m)
Moderate to strong
Moderate (on channel margins)
Slightly undulated topography
Late Pleistocene (140 to 70 ka)
Dendritic with variable, smooth, and flat microtopography
High (3–4 m)
Very low (on channel margins)
Highly undulated topography
Dissected alluvial fan
Older than late Pleistocene (>140 ka)
Dendritic; dissected surfaces with smooth and narrow divides and slopes leading to channels
Very high (>5 m)
None (surface destroyed) to strong (surface preserved)
None (surface destroyed) to strong (surface preserved)
Very low (on channel margins)
Ballena topography, about 50% of surface is preserved with convex slopes; eroded surface composed mostly of pebbles to cobbles with petracalcic coatings
 At regional or macrotopographic scales (>10 m planimetric observation length), the morphology of these fans is reflected by the following surface attributes: (1) depth of channel incision, (2) density of active channels, (3) pattern of drainage networks, and (4) topographic characteristics spanning several channel networks on the same geomorphic surface. The channels developed on older alluvial fans have dendritic patterns and significantly greater depths of incision compared to the channels on younger alluvial fans that have mostly distributary channel patterns (Table 1). The density of active channels on older fans is significantly lower than their density on younger fans. Generally, Holocene alluvial fan surfaces have greater roughness at microtopographic scales compared to late Pleistocene alluvial fans which are relatively smoother. At macrotopographic scales, however, the oldest fan surfaces in the study area are more dissected and have developed a more complex drainage pattern at the fan surface as compared to younger surfaces. These surface characteristics vary across the alluvial fans in the study area and are primarily controlled by long-term changes in far-field tectonic deformation and regional climatic variations that have collectively influenced base levels of the Colorado River, and overall rates of sediment accumulation and dissection throughout the Quaternary.
3 Materials and Methods Used to Develop Multiparameter Roughness Model
 The two principal data sources used in this study include a 1:5000-scale geomorphic map of alluvial stratigraphy (hereafter observed) prepared using traditional mapping techniques to provide observed data and bare-earth lidar topographic data at 1 m horizontal resolution with 0.5 m horizontal and 0.3 m vertical absolute accuracies, from the U.S. Army Geospatial Center Imagery Office, to quantify surface roughness by using multiple topographic geometries across alluvial fan surfaces.
3.1 Geomorphic Map Preparation
 The geomorphic map used to represent the observed data was created by applying traditional Quaternary geologic and geomorphic mapping methods that delineate alluvial landforms on the basis of several indices. These indices include tonal, textural, and topographic qualities, such as differences in surface color, degree of dissection and channel network development, and density of vegetation [e.g., Bull, 1991].
 Mapping was performed by identifying geomorphic features that were spatially distinct and discernible using georeferenced aerial imagery. Map unit contacts were rendered directly on base layers in a geographic information system (GIS) platform using Environmental Systems Research Institute's ArcMap® software package. Digitizing of unit boundaries was done at a fixed map scale of 1:5000 using 1 m resolution National Agricultural Imagery Program (NAIP) aerial photographs acquired in 2010. Mapped boundaries of alluvial fan units were then verified through both field observations and analysis of soil stratigraphy. Age designations to map units of this study are based on the classification system of Bacon et al. [2010a] for similar terrain near Yuma. Five map units (Qf1–Qf5), oldest to youngest, were observed in the study area (Figure 2b). A brief summary of each observed map unit's description is presented in Table 1.
3.2 Surface Roughness
 Surface morphology of alluvial fans can be defined by the frequency (number of occurrence) and magnitude (size) of topographic irregularities at a given scale of observation. We present a way to quantify surface roughness by analyzing multiple topographic geometries of slope, curvature (tangent), and aspect in different scales of observation. We used the standard deviation (SD) of slope, curvature, and aspect to quantify the irregularities on a given alluvial fan surface in terms of roughness. We defined the standard deviation of a variable as
where, σv = standard deviation of variable v within a defined observation window (i.e., 3 cell × 3 cell window), vi = value of the variable in cell i of the window, = mean value of the variable within the window, and n = total number of the cells in the window.
 Previous studies have used SD of slope [e.g., Frankel and Dolan, 2007] to calculate surface roughness. We argue that the SD of slope alone does not completely define roughness. Although the SD of slope provides information on the variation in the magnitude (i.e., relief) of roughness elements in the vertical dimension, it does not sufficiently describe the variation in curvature and orientation of roughness elements with respect to the horizontal dimension. In this regard, we propose that the integration of the SDs of slope, curvature, and aspect better characterizes surface roughness at a given scale of observation.
3.2.1 Multiscale Surface Roughness
 Alluvial fan surfaces have roughness elements that are reflected with different wavelengths in topographic data. The patterns of surface roughness are considered to be primarily a function of both magnitude (i.e., relief or amplitude) and frequency (number of occurrence) of topographic irregularities over an observational area (window size). For example, Figure 4 shows a schematic diagram of how surface roughness depends on the size of observation window or scale and the frequency and magnitude of microtopographic versus macrotopographic relief features. When microtopographic and macrotopographic relief features are compared over the same observation area (Figure 4a), significantly different frequencies and magnitudes of features are observed. Short-wavelength (<10 m) microtopographic relief features, such as boulders, plant mounds and plant scars, and depositional bar-and-swale features that form surface undulations, collectively have high frequency but low magnitude (<2 m relief). Long-wavelength (>10 m) features representing macrotopographic relief such as dissected terrain composed mostly of ridges and incised washes have low frequency but high magnitude (2–10 m relief). The roughness observed using a window size that closely matches the wavelength of microtopographic roughness elements is always smaller than the roughness observed using a window size scaled to the wavelength of macrotopographic roughness elements (Figure 4b). The roughness observed in small windows is dominated by microtopographic features, whereas the roughness observed in large windows results from a combination of both microtopographic and macrotopographic features. If the observation scale is larger than the wavelength of the largest topographic feature, the roughness remains constant, owing to the contribution from elements of all wavelengths. Accordingly, the relation between observation scale and roughness shows an initial, rapid increase in surface roughness corresponding to an increase in the size of the window over which the roughness is determined. The window size at which surface roughness no longer changes rapidly defines the dominant wavelengths of surface geometries observed in the topographic data (Figure 4b). Determination of the appropriate scale of observation, therefore, is necessary to accurately calculate surface roughness. We plot mean roughness values as a function of observational window size for each alluvial fan unit to determine the dominant wavelength of surface geometries. We then determine the frequency and magnitude of roughness computed at the dominant wavelengths to characterize the overall roughness of an alluvial fan surface.
3.2.2 Slope-, Curvature-, and Aspect-Based Surface Roughness
 The surface morphologies of alluvial fans reflect various time-dependent soil geomorphic and erosional and depositional processes that modify the surface following cessation of deposition. The principal processes include (1) weathering of deposits exposed at the surface and soil development; (2) degradation of depositional bar-and-swale topography and development of desert pavement; and (3) erosional and depositional processes that modify channel characteristics, such as drainage pattern, density, sinuosity, and channel incision. In general, the first two processes modify the slope and curvature of fan surfaces, and the third process controls the density of channels and aspects of channel walls. Young (Holocene) alluvial fans have highly irregular surface topography because of the presence of fresh sediment, distinct depositional and erosional features, and distributary patterns of highly sinuous channels. Intermediate aged alluvial fans (late Pleistocene) have relatively smooth surfaces because of moderately to strongly developed desert pavements and low density of distributary to dendritic channels that are relatively less sinuous. Old alluvial fans (middle to late Pleistocene) have dendritic patterns of low density, relatively straight, and more deeply incised channels that separate smooth preserved surfaces with limited or poorly to strongly developed desert pavements (Table 1). According to well-documented and accepted alluvial fan surface evolution models [e.g., Bull, 1991], the depositional (original) surface topography of Qf1–Qf4 alluvial fan units in the study area was generally similar to the topography reflected on the youngest Qf5 fan unit. It is apparent that the morphologies of these fan units observed today, therefore, are the result of time-dependent processes of erosion and weathering that have influenced the shape, size, and sinuosity of channels, development of desert pavements and soil, and ultimately the morphology of alluvial landforms.
 We argue that using at least three topographic parameters such as slope, curvature, and aspect more completely captures all physical elements of fan surface evolution because (1) the SDs of slope and curvature mainly measure the variability in relief and shape of surface features, and (2) the SD of aspect measures the directional patterns formed by the orientation, density, and sinuosity of channels that characterize fan surface morphology. Maps of these parameters were developed from a 1 m lidar digital elevation model (DEM) using a nine-cell (3cell × 3cell) moving window (Figure 5a) and the following algorithms:
where dz/dx and dz/dy are the rates of change of the surface elevation in x and y directions, respectively, L is the width of the cell, and Ci is the elevation of the cell at ith location (Figure 5a). The atan2 is the arctangent function whose value ranges from π to −π.
 Maps showing the SDs of slope, curvature, and aspect were developed using moving windows of sizes ranging from 3 × 3 m to 150 × 150 m (3 cell × 3 cell to 150 cell × 150 cell) from maps of slope, curvature, and aspect. The largest window size was selected based on the average wavelength of the largest macrotopographic features observed, which is the highly dissected ridge channel topography of the Qf1 unit having an average wavelength of 150 m. The topography was effectively captured by a 150 × 150 m moving window. The aspect value developed from equation ((6)) ranges from 0° to ±180° with an assignment of 0° for east, 90° for north,−90° for south, and ±180° for west azimuths (Figure 5b). If this model is used to calculate the SD of aspect, then aspect data in west quadrants (i.e., west sloping) may have a high SD value in error because of the discontinuity in azimuth assignment. To decrease this effect, three other aspect models were also derived (Figure 5b). The actual SD value of aspect for each moving window operation was then determined from the four SD maps of aspect using the following equation:
where σasp is the actual SD of aspect at a particular location (grid cell) and σasp1, σasp2, σasp3, and σasp4 are the SDs of the aspect at that location developed by the approach shown in Figure 5b. We observed that the SDs of aspect computed from the analysis of four aspect maps represent the topography better than the SDs of aspect computed from a single aspect map.
 Because this study is focused on the characterization of alluvial fans based on the roughness of preserved and intact areas of fan surfaces, topographic data associated with steeply incised channel walls and associated colluvial sideslopes were excluded from the surface roughness analysis. The exclusion was performed on the basis of a cutoff value of the slope (>3°) determined by overlaying a slope map on aerial imagery. The surface roughness of the excluded area was then determined from the interpolation of roughness from surrounding areas.
3.3 Combination of Roughness Maps Based on Fuzzy Logic Approach
 All roughness maps were first standardized to a common measurement scale and then combined using fuzzy operators [Zadeh, 1965]. Data standardization is needed because the DEM-derived slope-, curvature-, and aspect-based roughness values are independent from each other and are measured in different units. Slope values were calculated in degrees with respect to vertical direction, aspect values were calculated in degrees with respect to horizontal direction, and curvature values were calculated in 1/m. In addition, standardizing the data to a common scale allows comparisons among the data. All the roughness maps obtained at each scale of observation were standardized from 0 to 1 using the following linear function:
where rmin is the minimum roughness value of a roughness map, rmax is the maximum roughness value of a roughness map, and rsi is the output standardized value computed for a roughness value (ri) of ith grid cell.
 Five fuzzy operators [An et al., 1991; Bonham-Carter, 1994; Chung and Fabbri, 2001; Regmi et al., 2010] can be employed to combine standardized values of two or more input maps. These operators (1) fuzzy OR, (2) fuzzy AND, (3) fuzzy algebraic sum, (4) fuzzy algebraic product, and (5) fuzzy gamma can be expressed mathematically as
where ris is the standardized value for the ith map (i = 1, 2, …, n) at a particular location (grid cell) and rcs is the combined output value.
 Fuzzy OR and fuzzy AND operators are appropriate if one of the input maps best characterizes the roughness of a particular location. The fuzzy OR operator returns the maximum standardized value of the input maps occurring at that location, and fuzzy AND operator returns the minimum standardized value of the input maps occurring at that location. Use of these operators reduces effect of the dependency of one parameter on the other. If the combination of two or more input maps best characterizes the roughness of that location, the fuzzy algebraic sum, fuzzy algebraic product, and fuzzy gamma operators are appropriate. The fuzzy algebraic sum and fuzzy algebraic product consider all standardized values of the input maps occurring at that location and combine them based on the expressions shown above. The fuzzy gamma operator combines outputs obtained from fuzzy algebraic sum and fuzzy algebraic product based on a gamma value. The value of gamma ranges from 0 to 1. In the fuzzy gamma operation, when gamma is 1, the combination is the same as the fuzzy algebraic sum, and when gamma is 0, the combination equals the fuzzy algebraic product. Therefore, the appropriate choice of gamma produces output values that ensure a flexible compromise between effects of the values obtained from the fuzzy algebraic sum and the fuzzy algebraic product. In all fuzzy operations, the output values range from 0 to 1, where 0 represents the smoothest surface and 1 represents the roughest surface. Further details about fuzzy logic and fuzzy operators are discussed in Bonham-Carter  and Regmi et al. .
 The roughness values computed from slope and curvature can be expected to be higher around convex and concave areas of the landscape and lower in flat and planar areas, suggesting that these parameters are correlated in some cases. We used the fuzzy OR operator to combine the standardized roughness values derived from slope and curvature so that the effect of dependency of one parameter on the other is reduced. The output map was then combined with aspect-based standardized roughness map using fuzzy algebraic sum, fuzzy algebraic product, and fuzzy gamma operations. Eleven combined roughness maps were prepared with values of gamma ranging from 0 to 1.
3.4 Classification of Roughness and Comparison With Observed Geomorphic Map
 The individual and combined roughness maps were smoothed using a 20 × 100 m moving window (mean filter). The moving window was applied in eight directions with an aspect interval of 45° to produce eight surface roughness maps. We used a rectangular moving window rather than a square or a circular window because the shapes of alluvial fans in the study area are relatively narrow and elongated in the downstream direction. The rectangular shape captured a larger area of a fan unit by decreasing the edge effects of the surface roughness values associated with adjacent and different fan units. The eight maps of smoothed surface roughness were combined at each location (grid cell) by using the expressions
where and are the maximum and minimum of the mean roughness values at each location obtained by the smoothing of data along eight directions . The youngest fan unit (Qf5) having the highest roughness compared to surrounding older units was classified from the map, whereas older units (Qf4, Qf3, Qf2, and Qf1) that are mostly surrounded by the Qf5 unit were classified using map.
 All classified roughness maps were statistically compared with the observed geomorphic map. The agreements between the pairs of predicted and observed fan units were determined in percent by taking the ratio of area matched between the pair to the area of the observed unit in the pair. The overall goodness of agreement between the predicted and observed maps was also determined in percent by taking the ratio of total correctly predicted area to the total area of the observed map (Table 2).
Table 2. Comparison of Observed Versus Predicted Maps Based on Standard Deviation of Slope, Curvature, Aspect, and Combination of All Parameters
Predicted Versus Observed Alluvial Fan Unit
Combined Accuracy (%)
Slope-Based Accuracy (%)
Curvature-Based Accuracy (%)
Aspect-Based Accuracy (%)
Bold numbers represent the percentage of areas predicted correctly for each pair of predicted versus observed alluvial fan units. The agreements (accuracy) between the pairs of predicted and observed fan units were computed in percent by taking the ratio of area matched between the pair to the area of the observed unit in the pair. The overall accuracy was determined in percent by taking the ratio of total correctly predicted area to the total area of the observed map.
Overall Accuracy (%)
 Five prominent alluvial fan units (Qf1 to Qf5) were differentiated from the expert-based geomorphic mapping component of the study. The Qf2 and Qf5 units comprised the largest areas, whereas the Qf3 and Qf4 units comprised the least (Figure 2b). For each window size and for each variable included in this study, ~6,000,000 individual surface roughness values were derived from the lidar topographic data (Table 3). Roughness values from slopes steeper than 3° were excluded from the analysis. The excluded area consists mostly of colluvial sideslopes and channel walls formed along incised channels across Qf1 and Qf5 units and comprises ~12% of the study area. Anomalously high surface roughness values from >3° slopes were excluded so that the topographic elements of these slopes would not be included in the surface roughness analysis. We regard fully preserved or mostly intact and planar landform surfaces as crucial criteria for accurately differentiating alluvial fans based on surface roughness.
Table 3. Statistics of Surface Roughness Values Obtained for Different Aged Alluvial Fan Units
Roughness values at maximum frequency (Max. Freq.) and maximum frequency × magnitude (Max. Freq. × Mag.) were obtained visually from Figure 9 by looking at where the curves for frequency and product of frequency and magnitude are at their maximum and then projecting to the x axis to find the range of SD values.
 Plots of roughness versus window length indicate that all three parameters of slope, curvature, and aspect are good predictors of alluvial fan surface roughness (Figure 6). In these plots, fans of different ages have distinctive roughness curves that collectively show trends of decreasing surface roughness with increasing age. In addition, the roughness curves have steep slopes up to a point where the window length reaches 7 m. Beyond this point, the slope of each roughness curve rapidly drops and eventually becomes flat (Figure 6). The characteristics of each curve imply that the 7 m window length scales to the dominant wavelength of the microtopographic roughness elements and that the roughness values observed at larger window lengths are the effect of both microtopographic and macrotopographic roughness elements. Box and whisker plots (Figure 7) and frequency distribution curves (Figures 8 and 9) of roughness computed within a 7 × 7 m window also show that alluvial fan surfaces of different ages have distinctively different frequencies and magnitudes of surface roughness values derived from the three parameters of slope, curvature, and aspect.
 Among the eleven maps of combined roughness developed using the fuzzy gamma operation, the roughness map developed with a gamma value of 0.6 most closely predicted the observed alluvial fan map. The overall prediction accuracy of the combined roughness map is ~61%, while the prediction accuracies of slope-, curvature-, and aspect-based roughness maps are ~43%, ~47%, and ~53%, respectively (Figure 10 and Table 2). Comparisons based on prediction accuracies indicate that the combined roughness values best discriminate the surface characteristics of the observed alluvial fan units. The oldest Qf1 and Qf2 units and youngest Qf5 unit had the best match percentage (Figure 11 and Table 2). The reason for the inaccurate match between the observed and predicted areas of the Qf3 and Qf4 units can be attributed to: (1) similarities in slope and curvature characteristics of the Qf3 and Qf2 surfaces and (2) similarities in aspect characteristics of the Qf4 and Qf5 surfaces (Figure 6).
 If we consider slope-, curvature-, and aspect-based roughness observed in a 7 × 7 m moving window as a proxy for the magnitude (size) of surface roughness elements, then the magnitudes and frequencies (number of occurrence) of these elements on different aged surfaces are well characterized by power functions with rollovers (Figure 9). These power function curves (Figures 9a, 9c, and 9e) show that small roughness values occur more frequently on older alluvial fan surfaces and large roughness values occur more frequently on younger alluvial fan surfaces and thereby imply that low-magnitude topographic features dominate older fan surfaces, whereas high-magnitude topographic features dominate younger fan surfaces. Although our algorithms can detect distinctive surface roughness, the question remains as to what types of topographic features are contributing to the detected roughness of each alluvial fan unit. To answer this question, we considered that the overall roughness of an alluvial fan surface is a function of the frequency and magnitude of the topographic features; therefore, topographic features having high frequency and high magnitude dictate the overall roughness of an alluvial unit. Because of the lack of data on frequencies and magnitudes of topographic features on each alluvial fan surfaces, we used frequency and magnitude of modeled roughness values as their surrogates. The frequency-magnitude relationships (Figures 8 and 9) of the data were then evaluated to determine the values of roughness which contribute most to the overall roughness of each alluvial fan surface. The resulting roughness values were then overlaid on aerial imagery to determine the topographic features representing these values.
 Relations between frequency and magnitude have been used to explain processes in fluvial [e.g., Wolman and Miller, 1960] and hillslope [e.g., Guthrie and Evans, 2007] geomorphology. In fluvial geomorpholgy, the geomorphic work performed by a river has been computed as a product of the frequency and magnitude of flow, whereas in hillslope geomorphology, the work performed by landslides has been computed as the product of landslide frequency and magnitude (area). Although geomorphic forces and their rates of occurrence can be significantly different in fluvial and hillslope systems, we consider that the concept of geomorphic work is also applicable to characterizing surface roughness on alluvial fans, because the magnitudes of the topographic parameters of slope, curvature, and aspect are a reflection of both primary depositional and subsequent erosional and surface weathering processes. For example, smooth fan surfaces are the result of long-term geomorphic work by soil geomorphic processes, whereas rough fan surfaces reflect recent alluvial deposition that has not been significantly affected by these processes. We consider that roughness values which have high frequency and high products of frequency and magnitude contribute most to the overall surface roughness. Such values of roughness for alluvial fan surfaces of different ages (Table 3) were determined from the plots of roughness values in x axis against roughness frequency and frequency-magnitude product in y axis (Figure 9) by looking at where the curves for frequency and frequency-magnitude product are at their maximum and then projecting to the x axis to find the range of roughness values. For example, plots indicate that the geometric irregularities having SD of slope of 0.8°–2.5°, SD of curvature of 0.09–0.3, and SD of aspect of 80°–90° (shaded areas in Figure 9), which are frequently occurring and have high products of frequency and magnitude, contribute most to the overall surface roughness of the Qf5 unit. Results show little variation in slope-, curvature-, and aspect-derived roughness values that have the highest frequencies and highest products of frequency and roughness magnitude on individual fan units (Table 3). Visual inspection of roughness values which have high frequency and high products of frequency and magnitude was performed by overlaying the roughness maps on aerial imagery. By doing so, it was apparent that the roughness values coincided with bar-and-swale features, plant mounds, and plant scars on the Qf5, Qf4, and Qf3 units. In contrast, roughness values on the Qf2 and Qf1 units were associated primarily with shallow, incipient channels forming surface undulations that likely developed by long-term sheet wash erosional processes (Table 3).
 The results of our analysis show that SDs of slope, curvature, and aspect can be used as predictors of surface roughness to differentiate alluvial fan surfaces (Figure 6); however, there are variations among the distributions of the roughness values determined from slope-, curvature- and aspect-based curves with respect to the observation scale and alluvial fan age. For example, the slope-derived mean roughness values at small observation window lengths (3 m–11 m) are similar for the Qf1 and Qf2 units, but the curvature- and aspect-derived roughness values within the same window lengths are different (Figure 6). Furthermore, the frequency distributions of aspect-derived roughness are not similar to the frequency distributions of slope- and curvature-derived roughness values (Figure 8). But why do these roughness magnitude and frequency curves vary?
 A possible explanation to this question is that fan surfaces consist of features having diverse geometries and no single parameter capture the signature of all types of topographic geometries. For example, the SD of slope captures morphologic variation of a surface only in the vertical dimension (relief) and aspect captures the variation only in the horizontal dimension (azimuth), whereas curvature (tangent) captures the variation in the pole direction to the surface. The differences in the roughness curves, therefore, reflect the three-dimensional geometry of a typical alluvial fan surface. The roughness values for the Qf5 unit shown by all curves (Figure 6) suggest that the surface geometry of this unit has the highest magnitude of roughness in both the vertical and horizontal dimensions compared to the roughness magnitudes of Qf1–Qf3 surface geometries. The surface roughness of this young alluvial fan unit is a reflection of well-developed bar-and-swale microtopographic relief related to an extensive network of braided distributary channels. Comparison of the curves of slope- and curvature-derived roughness (Figure 6a) between the Qf5 and Qf4 units suggests that these units significantly differ in relief and curvature. In contrast, the aspect-derived roughness values (Figure 6c) indicate that Qf5 and Qf4 units have topographic features of almost similar aspect, which are probably related to similar channel characteristics. We observed in the field that both Qf5 and Qf4 units have distributary patterns of highly sinuous channels; however, Qf4 surfaces are more subdued (less microtopography) than Qf5 surfaces. Taken together, the Qf2, Qf3, and Qf4 units have roughness curves indicating that the magnitude of the topographic features in all dimensions decreases with surface age. This decrease in roughness is the result of soil geomorphic processes (e.g., degradation of bar-and-swale, development of soil and desert pavement) and evolution of channel morphology that has modified and smoothed the alluvial fan surface over time. Evolution of channels formed on fan surfaces from distributary to dendritic patterns decreases the density and the sinuosity of channels, which in turn reduces the surfaces roughness. The evolution of channels in the study area is probably the result of increased rate of fan surface dynamics due to tectonics and climate change induced increased erosion and channelization. When landscape equilibrium changes significantly because of either tectonic and/or climatic forcing, headward erosion and incision of relatively older fan units by tributary channels can erode well-developed desert pavements and soils and increase the overall surface roughness at macrotopographic scales. Such rejuvenated roughness is well reflected on the oldest Qf1 unit, which is heavily dissected and exhibits a ballena (fan remnant) landform shape (Figure 2c). In this study, we predicted only the relatively flat and planar preserved ridges of the Qf1 unit by excluding the steeper slopes associated with incised channels. The results suggest that the preserved and intact surfaces of the Qf1 unit have the lowest roughness values (Figures 6 and 7).
 Although the results show that fan surfaces of different ages have different roughnesses, the surface features that control roughness on alluvial fans with respect to age need to be identified to understand the evolution and dynamics of the fan surface. An analysis of maps showing roughness values having high frequency and high products of frequency and magnitude (Figures 8 and 9 and Table 3) overlain on aerial imagery shows that the roughness of young alluvial fans (Qf4 and Qf5) appears to be represented mostly by microtopographic relief in the form of bar-and-swale features, plant mounds, and plant scars. The roughness of older fans (Qf1 and Qf2) at microtopographic scales is defined mostly by well-developed desert pavements, a lack of vegetation, and only shallow surface undulations, which collectively represent long-term soil geomorphic processes that have smoothed surfaces and made them more uniform. The roughness of an intermediate-age fan unit (Qf3) is reflected by microtopographic relief consisting of weakly developed pavement and subdued bar-and-swale features (Table 3).
 Our analysis of alluvial fan roughness as a function of age shows that an accurate measurement of roughness is based principally on the size of the observation windows (Figure 6). By increasing the size of the observation window, the roughness value on all fan units rapidly increases up to a certain point where its rate of change decreases as a function of window size (Figures 4b and 6). The length of the window corresponding to this diminishing rate of change scales to the dominant wavelength of topographic features at microtopographic scale. As the size of the window length becomes larger, the surface roughness values for each unit become similar. This implies that larger window size may incorporate surface geometries of surrounding fan units, or that the window size exceeds the wavelength of the largest macrotopographic feature. Analysis of surface geometries detected in 1 m lidar topographic data (Figure 6) indicated that the microtopographic features in Yuma have a dominant wavelength of 7 m.
 The method of integrating the surface geometries of slope, curvature, and aspect used in this study, a fuzzy logic approach, is simple and reproducible. This is the first study to employ high-resolution digital topographic data to successfully predict and then test the surface characteristics of a suite of variable-age alluvial fan surfaces using multiparameter surface geometries. The results of this analysis suggest that the combined use of slope-, curvature-, and aspect-based topographic parameters predicted surface roughness of alluvial fans more accurately than a single topographic parameter (Table 2). The multiparameter approach resulted in a ~61% prediction accuracy for differentiating alluvial fan surfaces with respect to age when compared to an observed data set prepared by traditional geomorphic mapping techniques (Figure 11 and Table 2).
 We suspect that 39% failure rate of the model is related principally to both the exclusion of slopes >3° and the contrasting scales between map unit boundaries. The failure rate may also be related to lack of accounting for surface roughness variability on each map unit owing to particle size compositions and provenance of surface deposits, along with not excluding areas with anthropogenic surface disturbance in the form of roads and developed areas.
 The main objective of this study was to identify and then separate alluvial fans by surface age using modeled surface roughness. To characterize surface roughness and relate it to a time-dependent landform evolution model, we used preserved and intact surfaces with slopes <3°. Although the inclusion of slope <3° was sufficient to differentiate most of the low-relief alluvial fan features in the study by surface age, this threshold slope limited the overall characterization of relief on the older and highly dissected Qf1 unit. As a result, it is likely that the exclusion of slopes >3° complicated the comparisons because one of the major geomorphic elements that influenced the identification of the oldest Qf1 unit during the field-based mapping was the degree of dissection and relief, which the model did not account for.
 In terms of the geospatial comparison of the map data sets, we note that the boundaries of the observed map units were delineated at a fixed scale of 1:5000, which is more generalized compared to the 1 m cell resolution and pixelated map unit boundaries generated on the predicted map. Our field mapping did not distinguish small areas (<25 m2) of preserved and isolated older fan surfaces commonly present within the boundaries of younger alluvial fans because of the selected map scale. In contrast, the map of predicted roughness using the best smoothing window (20 × 100 m) found for this study distinguished fan units as small as ~5 m2. This shows that the observed and predicted maps were generalized at significantly different scales, which possibly contributed to a lack of agreement. Furthermore, the observed map is based on an expert's qualitative judgment and prior knowledge, whereas the predicted map is developed based on the features detectable in 1 m lidar topographic data without considering the uncertainties associated with the data itself.
 The primary source areas of alluvial deposits in the study area consist of sedimentary and volcanic rocks (Figure 1). These two rock types have been shown to have different rates of surface weathering and degradation on alluvial fans [McDonald, 1994], which can potentially influence roughness variability on alluvial fan surfaces of the same age. The model, however, assumes that the alluvial fans have uniform composition and does not recognize the role of lithology in alluvial fan surface roughness. Similarly, anthropogenic surface disturbance can include areas with very high roughness values in comparison to undisturbed areas. Such rejuvenated surface roughness can be observed in and around roads, buildings, and construction sites within the study area (Figures 2a and 10). The model does not consider the role of lithologic and anthropogenic factors in roughness calculation, and it is likely that by not accounting for lithologic differences and anthropogenic disturbances, the accuracy between the observed and predicted data was reduced. We believe that if the model was to be modified by considering the principal and ancillary factors presented above, the modified model would have a more accurate result. Nevertheless, the technique is promising for differentiating the relative surface age of alluvial fans under assumptions that the initial roughness of all fan units following cessation of deposition were similar and that differences in roughness between fan units are the result of the span of time during which soil and surface geomorphic process operated continuously and uniformly across surfaces that are underlain with similar lithological compositions.
 High-resolution lidar topographic data were used to quantify roughness of alluvial fan surfaces within the Sonoran Desert near Yuma, Arizona. Roughness elements representing microtopographic relief features were detected by calculating surface roughness as the standard deviation of slope, curvature, and aspect within a 7 × 7 m moving window across a 1 m resolution lidar-derived DEM. Results indicate that surface roughness calculated based on a multiparameter approach using the topographic geometries of slope, curvature, and aspect adequately differentiated the morphological expression of five alluvial fan surfaces of different ages. The approach had a predicted accuracy of 61% when compared to the same observed area that was characterized by traditional, field-based, and image analysis mapping techniques. This automated and geometric-based approach has limitations, because the combined roughness values represent a standardized quantity and because the approach is not suitable for mapping the relative age of extremely dissected alluvial fans that lack extensive areas of preserved and intact surfaces.
 The combination of all three topographic parameters provides dimensionless quantification of surface roughness, which in turn can be used to create a digital map showing the distribution of alluvial fan surfaces of varying ages. This approach affords possible applications to better understand fan stratigraphy and related soil geomorphic processes influenced by climactic change [e.g., McDonald et al., 2003; Bacon et al., 2010b], soil hydrology and land degradation [e.g., Romkens et al., 2002; Caldwell et al., 2008], and paleoenvironmental change [e.g., McAuliffe and McDonald, 2006; Pierre et al., 2012] by characterizing the magnitude and distribution of alluvial fan sedimentation and by quantifying the physical aspect of the landscape dynamics. The approach also has a potential application to better understand the slip rates along faults [e.g., Frankel and Dolan, 2007] by generating age-based geomorphic map of alluvial fans that could be used at regional reconnaissance scale to site specific local scale when numerical age is lacking.
 The multiparameter approach presented here is most applicable to large areas of desert terrain that have limited geomorphic landform data but which require some level of detail that is beyond that presented in standard soil and geologic maps. For example, this approach could enhance regional mapping of the distribution of alluvial fans and other desert landforms, where geospatial knowledge of particular geomorphic surfaces would enhance analysis of surface processes including dust emission, vegetation cover, and surface hydrology. In addition, application of the multiparameter approach to characterize surface roughness could be integrated with traditional image-based mapping to both increase the accuracy of identifying map units and decrease the time required for mapping large areas.
 We thank Tim Minor and the U.S. Army Geospatial Center Imagery Office with lidar data acquisition and the U.S. Army for granting permission and access to conduct field work. We are grateful to the Editors of JGR, M. Dühnforth, and an anonymous reviewer for their comments that greatly improved the manuscript. Research was funded by the U.S. Army Research Office (DAAD19-03-1-0159) with contributions from U.S. Army Research Office (W911NF-09-1-0256). The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of the U.S. government.