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Geomorphology 27 Ž^1999. 151–
Cosmogenic nuclides, topography, and the spatial variation of soil
depth
Arjun M. Heimsath a,), William E. Dietrich a^ , Kunihiko Nishiizumi b^ ,
Robert C. Finkel c
a (^) Department of Geology and Geophysics, Uni Õ ersity of California, Berkeley, CA 94720-4767, USA b (^) Space Sciences Lab, Uni Õ ersity of California, Berkeley, CA 94720, USA c (^) Center for Accelerator Mass Spectrometry, Lawrence Li Õ ermore National Laboratory, Li Õ ermore, CA 94550, USA
Received 7 November 1997; revised 18 March 1998; accepted 14 May 1998
Abstract
If the rate of bedrock conversion to a mobile layer of soil depends on the local thickness of soil, then hillslopes on uniform bedrock in a landscape approaching dynamic equilibrium should be mantled by a uniform thickness of soil. Conversely, if the depth of soil varies across an actively eroding landscape, then rates of soil production will also vary and, consequently the landscape will not be in morphologic equilibrium. The slow evolution of hillslopes relative to the tempo of climatic variations and tectonic adjustments would suggest that local morphologic disequilibrium may be expected in many landscapes. Here, we explore this issue of equilibrium landscapes through a previously developed model that predicts the spatial variation in thickness of soil as a consequence of the local balance between soil production and erosion. First, we confirm the assumption in the model that soil production varies inversely with the thickness of soil using two independent methods. One method uses the theoretical prediction that at local steady state Ž^ soil production equals removal , the depth of. soil should vary inversely with hillslope curvature. The second method relies on direct measurements of in situ produced concentrations of cosmogenic 10 Be and 26 Al in bedrock at the base of the soil column. For our study site in Northern California, the two methods agree and yield the expression that the rate of soil production declines exponentially with the thickness of soil from 0.077 mmryear with no soil mantle to 0.0077 mmryear under 1 m of soil. We then use this function of soil production in a coupled production and diffusive model of sediment transport to explore the controls on the spatial variation of the depth of soil on four separate spur ridges Ž^ noses. where we measured the data for the function of soil production. Model predictions are sensitive to boundary conditions, grid scale, and run time. Nonetheless, we found good agreement between predicted and observed depths of soil as long as we used the observed function of soil production. The four noses each have spatially varying curvature and, consequently, have varying depths of soil, implying morphologic disequilibrium. We suggest that our study site has been subjected to a wave of incision and varying intensities of erosion because of tectonic and climatic oscillations that have a frequency shorter than the morphologic response time of the landscape. q 1999 Elsevier Science B.V. All rights reserved.
Keywords: geomorphology; cosmogenic nuclides; soil production; erosion; landscape evolution
) (^) Corresponding author. Fax: q1-510r643-9980; E-mail: [email protected]
0169-555Xr 99 r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 5 5 5 X 9 8 0 0 0 9 5 - 6^ Ž^.
1. Introduction
Soil on hilly and mountainous landscapes is pro- duced from underlying bedrock by various mecha- nisms and is transported downslope primarily by the processes of mass wasting and overland flow. Even predominantly soil mantled landscapes are rarely blanketed uniformly with soil. Instead, bedrock often crops out in locally steep areas, soils are typically thin to absent on narrow ridge crests, and soil tends to accumulate to considerable depths in valleys Že.g., Young, 1963; Arnett, 1971; DeRose et al., 1991; Dietrich et al., 1995; Gessler et al., 1995. This^. spatial variation in the thickness of soil may hold important clues about the pace and relative uniform- ity of landscape evolution. Gilbert Ž^1877. Ž pp. 103–105. suggested that the rate of conversion of bedrock to a mobile surface layer Ž^ soil. is a function of the overlying thickness of the soil mantle. If this is the case, then the simple observation that the thickness of soil varies across a landscape indicates that the rate of bedrock conver- sion to soil varies. This implies that different rates of lowering occur across the landscape and that the landscape is not in what Gilbert and then Hack Ž 1960. called dynamic equilibrium. It is possible, however, that this is not the case. Ahnert Ž^1987. demonstrated that spatial variation in the thickness of soil could occur if the relationship between the rate of soil production and the thickness of soil were to vary spatially across the landscape. This condition, perhaps because of variations in the underlying bedrock, would result in thin soils forming on rock more resistant to soil production and thick soils developing on rock more readily converted to soil. Such a state could result in identical rates of lower- ing and a hillslope with constant topographic curva- ture. Hack Ž^1960. implied such a trade off between soil production and depth when he contrasted the slope morphology of quartzite versus shale in the Appalachians. The suggestion that the local thickness of soil affects the rate of bedrock conversion to soil, and hence the rate of supply of erodable debris, has developed into a central theme in geomorphology, thanks largely to the clarity with which Carson and Kirkby Ž^1972. wrote about the topic in their seminal book. They illustrated the Gilbert idea with a simple
cartoon of a function of soil production and sug- gested that landscapes tend to be in either a weather- ing-limited condition Žwhere the potential erosion is greater than the production of soil and the land strips to bedrock^. or a supply-limited state Žwhere rates of erosion do not exceed rates of potential soil produc- tion. In more recent numerical modeling of land-^. scapes, this theme has been highlighted Že.g., Ander- son and Humphrey, 1989^. as central to understand- ing the pace and form of landscape evolution. Tucker and Slingerland Ž^ 1994 , for example, conclude that. the persistence of distinct passive margin rift topog- raphy, such as the Great Dividing Range of Aus- tralia, owes its origin to the emergence of a weather- ing limited condition preventing the rapid spread of erosion across the landscape. Despite the central importance of the function of soil production to understanding landscape evolution Ž (^) e.g., Rosenbloom and Anderson, 1994 ,. little is known about it. As Cox Ž^1980. described, two basic hypotheses exist about the probable shape of the function of soil production. The simplest hypothesis suggests that soil production is greatest when bedrock is just exposed, decreases with increasing depth of soil, and assumes that this decline is exponential. This was assumed to emulate the decreasing effec- tiveness of mechanical processes, such as freeze-thaw Ž (^) Ahnert, 1967. or biogenic disturbance ŽDietrich et al., 1995.^. A more complicated function, first reasoned by Gilbert Ž^ 1877, pp. 103–105. for frost and solution processes, suggested that rates of soil production are greatest under some finite depth of soil, decline with depths greater than the optimum, and are also lower than the optimum under shallower soils and exposed bedrock. This ‘humped’ function supports the intu- ition that some limiting depth of soil is required for animal burrowing or vegetative rooting, as well as the fact that water rapidly moves off bare bedrock. As soil thickens beyond the optimal depth, rates of soil production would decline similar to the inverse exponential function. As Carson and Kirkby Ž^1972. and Dietrich et al. Ž^1995. point out, however, an unstable positive feedback exists for soils thinner than the optimal depth. Changes in rates of erosion would either cause the soil to thicken to a stable depth, equal to or greater than the depth at the optimum, or strip the hillside to bedrock. The result
Fig. 1b. Soil horizonation is typically limited in^. such regions and, therefore, such soils have received relatively little attention by pedologists. We write the mass conservation equation for the depth of soil, h , to represent the balance between the local rate of soil production, y^ Ž E e.^ rŽ E t ., and the
divergence of the sediment transport vector, q ˜ s, as,
E h E e
r s s y r r y r =s P q ˜ s Ž 1.
E t E t
where r (^) s and r (^) rare the bulk densities of soil and rock, respectively, and e is the elevation of the bedrock–soil interface in a bedrock-fixed coordinate system. Dissolution is not specifically modeled. In- stead, we account for dissolution effects by measur- ing the bulk densities of samples taken from the field. The intensity of chemical weathering of the bedrock undoubtedly affects the local rate of soil production. This effect is treated empirically as part of the function of soil production. The simplest law for the transport of hillslope sediment was first articulated by Davis Ž^1892. and
Gilbert Ž^1909. and states that sediment flux, q ˜ s, is
proportional to slope, = z , such that,
q ˜ ss y K = z Ž 2.
where K is equivalent to a diffusion coefficient with units of L^2 r t and z is the elevation of the ground surface. This diffusive transport law is most appro- priately applied to hillslopes where no erosion occurs by overland flow and shallow landsliding is rare or absent. Some field evidence exists to support the linear dependency of sediment flux on slope ŽMc- Kean et al., 1993 , and the diffusivity,^. K , can be estimated with various field measurements Že.g., Fer- nandes and Dietrich, 1997.^. Diffusive sediment
transport is widely assumed and is used in extensive applications of analytical and numerical models of landscape evolution ŽCulling, 1963; Kirkby, 1971; Armstrong, 1987; Anderson and Humphrey, 1989; Koons, 1989; Howard, 1994, 1997; Kooi and Beau- mont, 1994; Tucker and Slingerland, 1994; also see Ellis and Merritts, 1994. Dietrich et al.^.^ Ž^1995. and Reneau and Dietrich Ž^1991. use it to represent bio- genic transport and McKean et al. Ž^1993. found it applicable to soil creep. We substitute this transport law into Eq. Ž. 1 and solve for soil production:
E e r (^) s E h rs (^2)
s y y K = z Ž 3.
E t r (^) r E t rr
If the local depth of soil does not vary signifi- cantly over time Ž^ i.e., d h rd t s 0. then steady state conditions apply, where soil production is balanced by soil removal, and:
E e rs (^2)
s y K = z Ž 4.
E t rr
Eq. Ž. 4 states that if the local depth of soil is constant over time, then soil production is proportional to the negative of the topographic curvature, y= 2 z Ži.e., topographic divergence is negative with units of Ly^1 and, therefore, positive soil production occurs on divergent parts of the landscape while soil accumu- lates in the convergent regions. In convergent areas,^. d h rd t / 0 because of accumulation and Eq. Ž. 4 does not apply. The assumption of local steady-state depth of soil is central to both our methods, hence, our investigation focuses on divergent areas Ž^ ridges. where linear diffusive sediment transport processes are likely to be predominant.
Fig. 2. Ž .a Site location for Tennessee Valley, Marin County, CA. Regional topography is shown from part of the Point Bonita, CA USGS 7.5 min quadrangle with 25 m contour lines. The Golden Gate Bridge is directly east of the scale bar. The inset map is this field area shown with 10 m contour lines drawn from high resolution digital data Ž^ modified from Heimsath et al., 1997. Open triangles on the inset map. show sample locations for the cosmogenic nuclide samples. Large bedrock outcrops were sampled from the upper region in sub-basin 2 as well as from a ridge top near two of the surveyed noses. Nose surveys are located with the symbols Ž=, open square, black circle, and black triangle. and correspond to the upper-corner symbols on the individual maps of Figs. 5 and 9. The two nuclide samples at the base of sub-basins 1 and 2 are stream sediment samples. Ž .b Photograph of Tennessee Valley with sub-basins 1 and 2 labeled, large bedrock outcrop, TV-4 shown, and surveyed noses numbered corresponding to Figs. 5 and 9.
sides Kriging^ Že.g., minimum curvature, inverse dis- tance to a power, nearest neighbor, polynomial re- gression, and triangulation with linear interpolation^. resulted in grid artifacts that misrepresented, or even completely transformed, the real topography. We calculate curvature at a point using the eight nearest neighbors Ž^ e.g., Moore et al., 1993a,b. with an algorithm similar to the one used by Dietrich et al. Ž^ 1995. To illustrate our method, we focus on a. small, representative piece of a convex nose from one of our field surveys ŽFig. 3.^.^ = 2 z at the central grid node, a5 posted above the q, is calculated with the elevations, E (^) n , of the nearest eight grid nodes,
2 2 Ž^ E^^2 q^ E^^4 q^ E^^6 q^ E^^8.^ q^ Ž^ E^^1 q^ E^^3 q^ E^^7 q^ E^^9 .y^12 E^5 = z s (^2) 4 b
The subscript numbers correspond to the numbers posted above the symbols of the grid nodes and b is the width of the cell. = 2 z is calculated with Eq. Ž. 6 for every grid node on the landscape and is interpo- lated to yield curvature at the depth-sample loca- tions, shown for example by the black dot above node a5. A similar algorithm for calculating curvature uses the elevations of the nearest points of the topo- graphic survey Ž^ the small dots in Fig. 3. Such use of. real data rather than interpolated data is computa- tionally more involved and involves moving a fixed-size window equivalent conceptually to choos-^ Ž ing a grid scale^. to every measurement of depth and using all survey measurements within that window to calculate curvature. Because little difference exists in the calculated =^2 z between the methods we chose the grid-based algorithm for its computational sim- plicity and applicability to any grid-based landscape model Žsee Moore et al., 1991 for discussion of digital data sources. The data from the grid yield a^. best-fit surface to the local points. Because we are most interested in the trend of the topography, not in the centimeter scale heterogeneity captured by plac- ing the survey rod on small bumps or troughs on the ground surface this best-fit surface seems appropri- ate. Very good agreement exists between the grid for topography and the real survey points, with no obvi- ous grid artifacts.
Fig. 3. A representative area from a surveyed nose. Grid node locations from a 3.5-m grid interpolated from the original survey data are shown by q’s. Locations of survey points are shown by the small dots for this area. The large dot above grid-node 5 shows the location of one measurement of soil-pit depth. Contour lines are drawn at 1 m intervals. Curvature at every node is calculated using Eq. Ž. 6 in the text, written to illustrate the curvature calculation for grid-node 5. Curvature calculated in this manner is then interpolated from all adjacent grid nodes to determine the curvature at the location of the depth measurement.
One complicating factor in finding a robust method for calculating curvature is determining the appropriate scale of the topographic grid. We sur- veyed our field areas at very high resolutions to allow us to explore the effects of various sizes of grids Ž^ ranging from 1.5 m to 20 m. on calculated curvature. Fig. 4 shows an example of the effect of size of grid on curvature. For this example, curvature is plotted as a function of grid size for 21 points of depth on one of the surveyed noses. The curvature becomes relatively scale-independent at grid scales greater than 5 m. We applied such an analysis to all the measurements of depth for this study and deter- mined that a 5 m grid size was the finest size above which curvature was not sensitive to the scale of the grid. Grids with higher resolutions produce large local variances in curvature as the micro-topography approaches the scale of gopher mounds and animal trails. Conversely, sizes of grids greater than about
Fig. 4. An example of how grid size effects the calculated curvature for 21 measurements of depth from one of the noses shown in Fig. 5. Categories of grid sizes are 1.5, 2.5, 5, 7, and 9 m. Curvature becomes scale-independent at the 5 m grid size. We applied this method with grid sizes up to 20 m for all four of our surveyed noses shown in Fig. 5 with all measurements of depth and found that, in general, 5 m was the grid size that best characterized the land surface. Depth measurements with negative curvatures Ž^ concavity , or strongly scale dependent. curvatures were not used for determining the function of soil production.
10 m tend to smooth the landscape beyond the scale of the biota affecting the depth of soil in this field area Ž^ i.e., no large trees exist. The optimal scale of. the grid is likely to be different for landscapes under different dominant processes or climates.
4.2. Results
Four convex noses surveyed for this analysis are located in the study catchment by the symbols on the inset map of Fig. 2. These noses were chosen for the relatively high degree of curvature and for relatively gentle Ž^ - 258. slopes to avoid the possible influence of shallow landsliding. All four noses were surveyed with a total survey station at a 1–3 m resolution. Each nose is primarily underlain by greywacke sand- stone, although noses 2 and 4 had prominent out- crops of greenstone near the areas surveyed for this study. Fig. 5 illustrates the topography of these noses
with 2 m contour intervals. We intentionally sur- veyed divergent regions away from potential bound- ary affects such as the ridge crests and deep colluvial fills in the convergent hollows. An inverse relationship exists between curvature and the depth of soil on these noses Ž^ Fig. 6. which, if curvature is a surrogate for the production of soil, defines the form of the function of soil production. The symbols plotted on Fig. 6 correspond to the upper-corner symbol on each topographic map and suggest that each individual nose may define a slightly different relationship between soil produc- tion and depth. If this is indeed the case, and a single function of soil production exists for the underlying greywacke, then different noses may be producing soil, and, therefore, lowering at different rates. The variance in this curvature–depth relationship can be explained in two ways. The production of soil by biota may be thought of as stochastic processes acting on a landscape and causing significant short-
Fig. 6. Negative hillslope curvature divergent slopes are positive , Ž. y=^2 z , in my^1 , against measured local depth of soil, in cm, from the noses shown in Fig. 5. The symbols correspond to the upper-corner symbols on the individual noses. Curvature was calculated by the method illustrated in Fig. 3 and Eq. Ž. 6 in the text. Data plotted here retained the same curvature over the grid sizes and eliminate some of the points plotted on the maps because of insufficient survey points. We plot deeper depths here despite the weak concavities to show the strong inverse trend of the data over the range of the depths measured on Nose 3. Ž (^) Modified from Heimsath et al., 1997..
we interpret our observations of curvature and depth as showing a central tendency for thickness to in- crease with decreasing curvature.
5. Cosmogenic nuclides and soil depth
5.1. Method
As an independent test of the function of soil production determined from morphometry we use concentrations of cosmogenic nuclides in bedrock to infer long-term rates of erosion Žsee reviews in Lal, 1991; Nishiizumi et al., 1993; Bierman, 1994; Cer- ling and Craig, 1994. Our method relies upon mea-^. suring the concentrations of in situ produced cosmo- genic 10 Be Ž^ t s 1.5 = 106 y. and 26 Al^ Ž t s 1 r 2 1 r 2 7.01 = 10 5 y. extracted from the target mineral quartz in bedrock ŽLal and Arnold, 1985; Nishiizumi
et al., 1986; Lal, 1988; Lal, 1991; Nishiizumi et al.,
- If we assume that the conversion of bedrock^. to soil reaches a steady state under a constant depth of soil, h , Žand the soil bulk density remains con- stant^. then, the concentration of the cosmogenic radionuclide, C^ Ž^ atomrg ,. in the bedrock at the soil-bedrock interface is,
C s P Ž h , u. r ´r Ž 7.
l q 0
L
where P h^ Ž^ , u. is the rate of production of the nuclide Ž^ atomry. at depth h and slope u , L is the mean length of attenuation Ž^ ; 165 grcm^2 ., l is the decay constant of the radionuclide and Žl s ln 2rt (^1) r 2 .and ´ is the rate of conversion of bedrock to soil Ž^ cmry. Ž i.e., yE e rE t in Eq. Ž .. 1. Eq. Ž. 7 is of the same form as that used by others to calculate the rate of erosion of bedrock Žin which case, h and u s 0. Ž Lal and Arnold, 1985; Nishiizumi et al.,
- We can rearrange Eq.^.^ Ž. 7 to solve for the conversion rate of rock to soil as a function of either measurable or known quantities such that,
E e L P Ž h , u.
´ s y s ž / Ž 8.
E t r (^) r C y l
The rates of production of 10 Be and 26 Al in quartz are known at the ground surface as functions of latitude, elevation and topographic shielding ŽLal, 1991; Nishiizumi et al., 1989. We measured the^. bulk density and depth of soil individually. Most of our samples are taken from hillsides with slopes between 10 and 30 8 , which effects the relative expo- sure to cosmic ray flux. Nishiizumi et al. Ž^1989. discuss the implications of sampling from inclined surfaces, referring to the original observation of Lal Ž 1958. that the angular distribution of cosmic ray particles in the troposphere is given by F^ Ž^ u .s sin 2.3 u. This simple correction for slope is combined with a depth correction ŽNishiizumi et al., 1991, Eq. Ž .. 1 for the rate of production of nuclides when the bedrock sample is under a depth of soil and the soil mass shields the bedrock from the penetration of cosmic rays Ž^ Lal, pers. comm. 1996. We use the soil.
bulk density and assumed steady-state soil depths Ž (^) Table 1. to calculate the shielding mass used for the depth correction. We sampled bedrock in soil pits from the continu- ous layer of rock at the soil–bedrock boundary and from exposed bedrock Ž^ the h s 0 samples. Sam-. pling pits extended below this boundary to insure accurate local measurement of the boundary. We excavated the top 1–3 cm of bedrock from the in-place bedrock with a chisel and hammer. About 1–3 kg of bedrock were required to insure 40–60 g of quartz remained after rock crushing and chemical separations. Chemical separation and purification of quartz from the sampled rock followed the procedure outlined by Kohl and Nishiizumi Ž^ 1992. We used. four 10-h ultrasonic leaches, however, compared to the normal three leaches that are typically sufficient for ‘cleaner’ quartz. This insured removal of any meteoric Ž^ garden variety. 10 Be remaining in the sam- ple and helped reduce the background concentration of 27 Al. The quartz contained high levels of titanium, which can potentially scavenge Al from the analysis
solution by co-precipitation. Each precipitate was, therefore, treated at least twice with sulfuric acid. No significant residue Ž^ - 1 mg. remained after this treatment and we found no evidence of scavenged Al. We measured background concentrations of 27 Al by AA. We used a Be carrier calibrated by Nishi- izumi’s Be AA standard and it differed by less than 2% from the Be carrier used for the Nishiizumi et al. Ž 1989. analyses. (^10) Be and 26 Al concentrations were measured at the LLNL-CAMS facility Ž^ Davis et al., 1990. and the measured ratios were normalized to the ICN 10 Be and the NBS 26 Al standards. Rates of soil production rates were calculated from these concentrations using Eq. Ž. 8 and the depth-slope correction factor for samples from under the full range of soil depths. We quantified the function of soil production by plotting these rates against the measured depth of soil. As a separate measure of the average rates of soil erosion from the hillsides, we sampled stream sediments from sub-catchment outlets ŽBierman and Steig, 1996; Granger et al., 1996. We also sampled three^.
Table 1 Measurements of cosmogenic nuclide concentrations Sample Depth Slope Elevation Quartz wt. 26 Al 10 Be 26 Alr^10 Be h -slope yd e rd t Ž (^) cm. Ž (^) deg. Ž (^) m. Ž .g Ž (^10 6) atomrg. Ž (^106) atomrg. factor Ž (^) mrMy.
TV-2 0 8 135 40.65 0.600 " 0.101 0.115 " 0.006 5.22 " 0.92 1 39 " 8 TV-3 16 10 120 40.06 0.447 " 0.054 0.108 " 0.007 4.15 " 0.56 0.87 47 " 15 TV-4 0 15 275 42.68 1.132 " 0.078 0.248 " 0.010 4.55 " 0.37 0.98 20 " 5 TV-5 0 0 275 40.22 1.433 " 0.121 0.351 " 0.017 4.08 " 0.40 1.00 15 " 4 TV-6 35 15 105 42.03 1.104 " 0.050 0.195 " 0.008 5.65 " 0.34 0.69 26 " 3 TV-7 58 20 100 42.58 1.446 " 0.061 0.229 " 0.016 6.32 " 0.52 0.52 21 " 3 TV-10 51 17 115 40.71 0.939 " 0.104 0.171 " 0.011 5.49 " 0.71 0.59 25 " 4 TV-11 0 21 120 32.39 0.234 " 0.035 0.040 " 0.005 5.85 " 1.20 0.98 107 " 23 TV-12 30 15 116 40.56 0.394 " 0.091 0.074 " 0.006 5.28 " 1.30 0.72 60 " 16 TV-13 49 18 140 40.03 1.060 " 0.112 0.151 " 0.013 7.03 " 0.95 0.61 26 " 5 TV-15 20 15 135 40.33 0.563 " 0.055 0.083 " 0.006 6.77 " 0.83 0.85 48 " 8 TV-16 35 20 133 45.05 0.741 " 0.076 0.134 " 0.007 5.54 " 0.64 0.68 33 " 5 TV-17 60 25 133 52.57 0.914 " 0.093 0.161 " 0.015 5.68 " 0.77 0.54 27 " 5 TV-23 0 15 137 40.28 0.260 " 0.051 0.050 " 0.005 5.16 " 1.14 0.98 91 " 24 creek1a nra nra 110 51.50 0.414 " 0.053 0.063 " 0.005 6.61 " 1.00 1 64 " 12 creek1b nra nra 110 40.22 0.366 " 0.050 0.063 " 0.006 5.81 " 0.95 1 66 " 13 creek2 nra nra 110 58.22 0.239 " 0.049 0.041 " 0.005 5.87 " 1.37 1 102 " 25
Concentration errors include 1 s from AMS. All errors propagated to yd e rd t. Average soil density: 1.4 grcm 3 Location: 37.9 N Lat., 122.6 W Long. (^26) Al and 10 Be production rates are corrected for elevation and location Ž (^) Lal, 1991; Nishiizumi et al., 1989 ..
h -slope factor corrects for soil depth and slope shielding Ž^ Lal, pers. comm. ..
from Fig. 6, to rates of soil production according to Eq. Ž. 4 using an independently determined regional diffusivity of 50 cm 2 ry Ž^ data from Reneau Ž^ 1988 ,. reported in McKean et al. Ž^ 1993 , and used in Diet-. rich et al. Ž^1995 .. , and an average bulk density ratio of 0.5. Fig. 8 shows a direct overlay of these values of soil production on the nuclide-based results plot- ted in Fig. 7. We emphasize that no parameters were adjusted to compare these data from the two com- pletely independent methods. The similarity between these two data sets supports considerably these two approaches. This suggests that we should be able to use Eq. Ž. 9 in a numerical model to predict the local soil depths.
6. Numerical modeling of soil thickness
6.1. Method
We used our function of soil production to predict the thickness of soil by adapting the model proposed by Dietrich et al. Ž^ 1995. The model solves Eq..^ Ž. 1 by finite differences, adjusting soil production as a function of the local soil depth at each time step. The surface topography evolves such that after each time step the local flux and rates of soil production are recalculated. Here, Eq. Ž. 9 was used for the function of soil production and the model was applied to real topography from the four individual noses shown in Fig. 5. The model was run with 10-year time inter- vals using the smallest grid Ž^ 1.5 m. that closely fit the survey data for the noses. Sediment is transported to and from the eight nearest grid cells as a function of gradient with a 6 2 correction applied to the diago- nals to account for the greater length of transport. If depth of soil thins to zero at a grid node, and downslope gradient demands more sediment than is available, then all sediment produced and received from above is transported out. We determined the total time that the model was run, in part, by plotting the predicted depths of soil against time and identify- ing the time at which the rate of soil depth increase declined significantly. This point was chosen to ap- proximate the time when the local depth of soil had reached steady state and soil removal balanced soil production.
We needed to specify three boundary conditions to change the application of the model from the catchment scale Ž^ Dietrich et al., 1995. to the nose scale. At the upslope boundary of the noses, we set the maximum possible influx of sediment equal to the maximum rate of soil production of 77 mrMy. This is equivalent to the weathering-limited condi- tion of exposed rock such that the upper bound of the noses can only be supplied with sediment that is available from the single row of boundary grid cells. We chose this condition to represent the thin soil and exposed bedrock near the crests of noses 1 and 3 and the upper end of noses 2 and 4 Žthe upper boundaries of our surveyed regions. Boundaries along the sides^. of the surveyed noses were set to remove any sedi- ment transported to them to avoid the accumulation of sediment. These side boundaries represent a con- dition of steady state elevation where an exact bal- ance exists between inflow and outflow of sediment, and attempts to represent the position of the study areas Ž^ where depths were measured. well above the sediment accumulation zone of the hollows. The lowermost boundary is set to lower at a rate equal to the rate of soil production under the field-observed average depth of soil for the lower boundary of the respective nose. This condition maintains the ob- served landscape position of our surveyed noses above the sediment accumulation zone near the val- ley bottoms and, therefore, reduces flattening of the topography during the model runs. We specified zero depth of soil across each of the noses for our initial condition for the results reported here. As a means of comparison, we also ran the model with an initial depth of soil of 30 cm.
6.2. Results
The model predicts the thickness of soil to be thinnest on the nose axes and to be locally highly variable Ž^ Fig. 9. In general, thickness of soil in-. creases downslope away from the nose crests. The very thin soils at the top of each nose are an artifact of the upslope boundary condition. We avoided this artifact in our comparison of observed with the predicted depths of soil by using only the observa- tions below the highlighted contour lines of Fig. 9. Regions of thicker soils correspond to weakly diver- gent or slightly convergent areas on the noses. In a
Fig. 9. Predicted depths of soil, in m, after 6000 years of running the soil model with 10-year time steps on all four surveyed noses. These predicted depths are determined using the exponential fit to the data here, shown in Fig. 7, as the function of soil production. Boundary conditions are explained in the text and initial soil thickness is zero. Topographic contour lines are drawn at 2 m, and are the same as Fig. 5. The thickened contour line near the top of each nose shows the cut-off elevation above which we treated predicted depths as upper-boundary artifacts and did not compare them to our observed depths in the area.
shows a much larger range of depths over the range of curvature measurements as shown in Fig. 6, and the deeper soils were found in weakly convergent areas. Our conceptual model applies the assumption of a steady state depth of soil only to the divergent areas of the landscape and the deeper soils that we observe may result from, in part, accumulation in locally convergent areas.
7. Discussion
7.1. Steady-state soil depth
An important assumption that is common to both methods is that the local thickness of soil does not vary with time as soil is produced and transported downhill. This assumption of local steady state is justified at our field sites in several ways. No evi- dence exists for shallow landsliding or erosion by overland flow on the convex regions that we studied. Soil production, primarily by burrowing mammals, while stochastic, tends not to alter local soil thick- ness dramatically beyond a few years. Numerical experiments by Dietrich et al. Ž^1995. show that on ridges eroding by diffusive processes, initial arbitrary thickness of soil quickly Ž^ in a few thousand years. reaches a local steady-state for a slowly changing curvature. Our modeling results Žusing much higher resolution topographic data^. show that thickness of soil quickly adjusted to local curvature, but because of the boundary conditions and the initial convexity of the nose, curvature slowly changed with time thus causing the local thickness of soil to change. The greater sensitivity to curvature change in our model- ing results from the higher resolution topography that we used, which led to larger local variations in elevation and, therefore, curvature. Also, the initial surface used in our model was effectively much rougher and, therefore, smoothed more than the larger scale modeling reported by Dietrich et al. Ž^ 1995 .. Because of the rapid thickness of soil adjustment to the initial thickness imposed on the noses observed in our models, we suggest that any adjustment in soil thickness in response to Holocene warming and dry- ing occurred in the early Holocene. The concentrations of 10 Be and 26 Al provide an independent test for our assumption of steady-state
depth of soil. 10 Be has a half-life roughly twice as long as 26 Al and the ratio of 26 Alr^10 Be can be used to help infer the history of erosion and exposure for the samples ŽNishiizumi et al., 1993; Nishiizumi et al., 1991. Nishiizumi et al.^.^ Ž^1991. illustrate how this measured ratio can be used for such conclusions. The short exposure ages of our samples, resulting from relatively high rates of erosion, means that all our samples should have a 26 Alr^10 Be value of about six Ž (^) Table 1 .. A few of our samples show some discrepancy between the measured 26 Alr^10 Be ratios and the ex- pected value of 6.0 1991 Ž^ Nishiizumi et al., 1991 .. We were concerned that the difference in the mea- sured 26 Alr^10 Be would significantly affect the nu- clide-determined function of soil production. When we plot the function of soil production Žin the same manner as Fig. 7^. using only the concentrations of (^10) Be or 26 Al to calculate the rates of soil production we find the following best-fit lines regressed by standard error-weighted least squares ŽBevington, 1969 :^.^ Ž. 1 using 10 Be only: soil productions Ž 74 "
- = eŽ y0.022^ "^ 0.002^ )^ depth.^ , Ž. 2 using 26 Al only: soil productions Ž^79 " 7. = eŽ y0.024^ "^ 0.002^ )^ depth., and Ž. 3 using 26 Al and 10 Be: soil productions Ž (^77) " 9. = e^ Ž y0.023"^ 0.002^ )^ depth.^ Žthe fit shown in Figs. 7, 8 and 11.^. These functions of soil production are statistically the same. We chose to use the average measurements of (^10) Be and 26 Al in our calculation of the function of soil production because we see no independent rea- son to reject either the 26 Al or the 10 Be data. Three samples differ unequivocally from 6, notable the samples from large bedrock outcrops, TV-4 and TV-5, and the shallow soil mantled, TV-3. Only TV-3 is used in the regression that defines the function of soil production, but its influence on the slope of the line is small because of its larger variance. Samples TV-4 and TV-5 require further discussion. These low 26 Alr^10 Be ratios could only arise if, Ž. 1 some systematic error occurred in sample analysis Ži.e., overestimation of the concentration of (^10) Be from incomplete removal of garden variety 10Be, or underestimation of 26 Al from incomplete dissolution of the sample or errors in AA measure- ment of the concentration of stable Al in the sample solution^. or 2. some complex burial history occurred
Fig. 11. The exponential best-fit to these data in Fig. 7 plotted with functions of soil production used by Dietrich et al. Ž 1995 .. We use a linear axis for the rate of soil production here to better illustrate the polynomial function plotted with the solid grey line. The best fit to our data is plotted with the small black dashed line with an intercept of 77 mrMy and a slope of y0.024. The exponential function used by Dietrich et al. Ž^1995. is plotted with a large dashed line, an intercept of 190 mrMy, and a slope of y0.05. The large black dots are the data from Fig. 7 used to derive the exponential best fit for this function of soil production. The convergence of all three curves occurs at about 30 cm of depth in the soil.
during which time the 26 Al decayed faster than the (^10) Be. No analytical reason exists to reject the analy-
ses of 10 Be or the 26 Al. Interestingly, TV-4 and TV-5 are large bedrock outcrops, lithologically different from the other sam- ples, which are physically separated from the other samples by lying at the top of the basin ŽFig. 2a inset map and labeled on Fig. 2b. The low ratios could^. point to some complex geologic history that will require further analysis to be fully understood. Here we choose to interpret these samples with our simple exposure model, and express the difference between the results of 10 Be and 26 Al with the large uncer- tainty in the estimates of erosion for these samples. The only conclusion that we draw from these sam- ples, and TV-2, is that large bedrock outcrops are eroding, or lowering, at significantly slower rates that the soil-mantled part of the landscape. The
uncertainty between the 10 Be and the 26 Al based results does not alter this conclusion.
7.2. Soil depth and cur Õ ature
The curvature–depth analysis depends on using an appropriate spatial scale to calculate curvature and on correctly identifying the soil–bedrock bound- ary. Several reasons exist to explain why this analy- sis may break down. The first is that the identifica- tion of the boundary may be incorrect because large pieces of rock in the colluvium are very similar to the fractured bedrock. We found that a soil auger often could not get past colluvial stones and would, therefore, not reach the soil-bedrock boundary. The presence of stone-lines would have further compli- cated our measurements. In addition to relying on soil pits for our measurements we extended the pits beyond the soil–bedrock boundary to insure that the observed fractured bedrock was not a stone-line. Even if the actual identification of the soil–bedrock boundary was correct, the observed depth of soil may not reflect the long-term average depth. If the dominant process of soil production-transport had changed recently because of land-use or climate changes, then the depth of soil may be still adjusting toward a local steady state in response to the new processes. For example, if the mid-Holocene climate in this area was significantly drier ŽRypins et al.,
- then rates of soil production from biotic activ- ity may have slowed in comparison to the rates in wetter conditions. The current depth–curvature rela- tionship could still be adjusting from such a mid-Ho- locene condition. Numerical modeling, however, in- dicates relatively rapid adjustment of the local depth of soil, and suggests that this probably is not the case. Fernandes and Dietrich Ž^1997. solve a diffusion- type equation in one-dimension Ž1-D equivalent to our Eq. Ž. 4 written for the change in elevation of the ground surface rather than the soil–bedrock inter- face^. with parameters specific to this field area here to explore the equilibrium condition of convex hill- tops. Their analysis of morphologic relaxation time for hillslope profiles after just a twofold change in either diffusivity or the rate of base-level downcut- ting suggests that the time to morphologic equilib- rium is on the order of seventy thousand years or
The exponential function of soil production shows the highest rate of soil production occurring under no soil cover. If burrowing gophers and the penetration of roots from vegetation are the primary agents of mechanical disruption of the bedrock, then it may be that some limited soil mantle is necessary for the highest rate of soil production. On the other hand, the bedrock that emerges at the ground surface is typically highly fractured and friable, and probably undergoes accelerated breakdown as a consequence of wetting and drying. Periodic fires may also con- tribute to the rapid breakdown of exposed rock. Without further investigation into the mechanisms of soil production we cannot reject the possibility of a ‘humped’ function of soil production, but the evi- dence reported here strongly supports a simple expo- nential function.
7.3. Landscape equilibrium
A variety of observations, in addition to the depth–curvature relationship, suggest that this study site is not in morphologic equilibrium. Large bedrock outcrops are eroding at rates significantly lower than the rest of the landscape. The largest outcrop ana- lyzed, TV-4, sticks up above the surrounding soil by an average of about 10 m. This difference in eleva- tion would develop over about 270 000 years if TV-4, which is eroding at about 40 mrMy, emerged from a surrounding landscape lowering at the maxi- mum rate of 77 mrMy. This is a long term, local perturbation of the topography. Furthermore, the (^10) Ber (^26) Al ratio of this and other large outcrops
suggests that a complex history of burial and expo- sure occurred here. Smaller outcrops are scattered across the hillslopes and can cause local topographic perturbations that may persist for millennia after the outcrop is stripped away by erosion. The average rates erosion from the two subcatch- ments in the steeper areas of the headwaters ŽFig. 2a inset and b^. are higher than the rates of soil produc- tion on the noses in the gentler slopes of the lower part of the study basin. The only nose in the steeper part of the catchment that we surveyed ŽNose 2, shown by the X in Fig. 2a inset and labeled on 2b^. had distinctly thinner soils and lacked the areas of low curvature Žand, therefore, a low rate of soil production^. found on other noses. Additionally, most
of the shallow landslides mapped in this study area occurred in the steeper part of the basin ŽDietrich et al., 1993. These observations together suggest that^. the lower, more gently sloping parts of the study area are eroding distinctly more slowly than the steeper headwaters region. More speculatively, but consis- tent with these observations, the surrounding ridge of the catchment is sloping relatively gently. We sug- gest that a wave of incision moved up the catchment, with the lower region in a state of ‘relaxation’ Že.g., Ahnert, 1987 , or slowing rate of erosion.^. The tectonic setting of this study site is complex. To the east and west major active strike-slip fault systems exist. Although this site is east of the San Andreas Fault, it is part of a transpressional zone, components of which are advecting northwards Že.g., Aydin and Page, 1984; Prescott and Yu, 1986; Page,
- San Francisco Bay^.^ Ž^ 4 km south of the site. is subsiding at 0.07–0.8 mmryear Že.g., Atwater et al., 1977; Prims and Furlong, 1995^. whereas Quaternary marine terraces rise progressively higher north of the site Ž^ e.g., Wehmiller et al., 1977. Given this setting,. it is extremely unlikely that tectonic-induced river incision has been constant over the time scale neces- sary for morphologic equilibrium of the landscape to develop. Climatic variations in the Quaternary have also contributed to the disequilibrium here. A Holocene alluvial and colluvial fill occurs in the main valley floor of the study site as well as in most of the valley network ŽMontgomery and Dietrich, 1989; Dietrich et al., 1993. The lowering of sea level induced by^. glaciation may have caused periodic channel incision by lowering the base-level Žthe Pacific Ocean is currently less than 2 km west of the study site.^. Higher rates of rainfall and runoff during the Pleis- tocene also may have led to periods of active chan- nel incision. If these periods of increased channel incision did occur, then the base of the hillslopes were subjected to periodic steepening that would have then advanced upslope, leading to long periods of morphologic adjustment. The curvature variation that we observe suggests disequilibrium in the morphology of the landscape and it leads to the spatial variation in soil thickness. Curvature variation is not randomly distributed. The axes of all four surveyed noses are systematically the most strongly curved part of the noses. This curva-
ture is nearly all plan curvature rather than profile curvature. It is possible that this spatial variation of curvature is an expression of nonlinear processes of diffusion sediment transport ŽRoering et al., 1997, in press. Even so, the spatial variation of soil thickness^. associated with the curvature would still imply non- uniform rates of soil production. Given the modest gradients of these study sites, we believe that an approximation of linear diffusion is applicable and that systematic flattening away from the nose axes may have resulted from a reduced rate of incision in adjacent valleys. We are exploring the possibility of nonlinear sediment transport through further model- ing and field work.
8. Conclusion
We suggest a conceptual framework for predict- ing soil thickness on a real landscape using a pro- cess-based model. This study supports the frame- work of Dietrich et al. Ž^1995. and uses our ŽHeim- sath et al., 1997^. field-derived function of soil pro- duction. Our quantitative determination of the func- tion of soil production has enabled reasonably accu- rate prediction of spatial variations in soil thickness on fine-scale topographic noses. We find, however, that predicting soil thickness depends on the bound- ary and initial conditions, the grid size, the model run time, as well as the function of soil production. The modeling results presented here to predict the thickness of soil help verify a method that could be applied at other sites to help understand the impacts of lithology, climate, and tectonics on landscape evolution. Additionally, such methods could be ap- plied to evaluate potential impacts of land-use in regions where soil thickness may be sensitive to changes in the dominant geomorphic processes by identifying how soil thickness could change when, for example, the diffusivity changes with changing strategies of land-use. The variable thickness of soil that we observe is a function of topographic curvature and suggests an inverse relationship between soil production and depth of soil. This relationship, along with several other observations about rates of erosion and land- scape form, suggest that the landscape is not in morphologic equilibrium. This disequilibrium is not
surprising given the complex tectonic setting of the study area and the likely residual influence of an oscillating climate on a landscape undergoing rela- tively modest rates of erosion. We do not, however, know the exact cause of the systematic variation in curvature observed here. Extension of these methods to new field sites will require very high-resolution topographic surveys and will benefit from higher spatial resolution of depth measurement than we used here. More densely spaced measurements of soil depth will enable accurate depiction of the bedrock surface, which can then be tested as an initial condition in modeling the devel- opment of the soil mantle. The topographic surveys should also include the complete topography, from ridge tops to valley bottoms, which can then be used to help clearly define the boundary conditions. Fur- ther application of the nuclide method would benefit from quartz-rich underlying bedrock and both meth- ods depend on being able to clearly identify the soil–bedrock interface. On landscapes undergoing very high rates of erosion, the nuclide method may become impractical if the concentrations of the nu- clides are below measurable levels. It may also be that landscapes, where the linear diffusive sediment transport law is applicable, may be limited and in- stead other processes of mass wasting, such as lands- liding, may predominate. Despite such potential limi- tations, this approach has empirically quantified the function of soil production for the first time and should be broadly applicable to other landscapes.
Acknowledgements We thank K. Heimsath and L. Cossey for field and lab assistance, D. DePaolo for laboratory space, D. Lal for suggestions and the Golden Gate National Recreation Area for access to our study site. D. Bellugi adapted the numerical model to conditions here and conducted the modeling efforts. This work was partially performed under the auspices of the US DOE by Lawrence Livermore National Laboratory under contract W-7405-Eng-48 and was supported by Cal Space, IGPP-LLNL GS96-05, NSF EAR- 9527006, with NASA Global Change and Switzer Environmental fellowships to AMH. We thank Gre- gory Hancock and an anonymous reviewer for their helpful suggestions.
