ied within the context of the diffusion model of

along the entire scarp pro¬le, and it is the point

hillslope evolution for over forty years. In that

that is also most sensitive to uncertainty in the

time, researchers have taken several different ap-

initial scarp angle. As a scarp evolves, the greatest

proaches to analyzing scarps. In this section, we

amount of erosion and deposition occurs where

present the analytic solutions to the diffusion

the magnitude of the hillslope curvature is great-

equation appropriate for pluvial shoreline and

est, near the top and bottom of the scarp. The

fault scarps. In addition, however, we also com-

scarp midpoint is a point of in¬‚ection with zero

pare and contrast past approaches in order to

curvature, and hence this point changes the least

put the analytic solutions into the historical con-

of all points along the pro¬le for young scarps.

text of other approaches. By comparing different

An alternative approach is to ¬t the entire scarp

methods for scarp analysis and modeling, we are

(or its derivative, the hillslope gradient) to an-

able to explore some of the subtleties involved in

alytical or numerical solutions of the diffusion

comparing model predictions with natural land-

equation (linear or nonlinear). Avouac and his col-

forms. For example, how do we handle uncer-

leagues (Avouac, 1993; Avouac and Peltzer, 1993)

tainty in the landform initial condition at some

pioneered the use of this technique in fault-scarp

point in the past? How do we handle measure-

studies in Asia. Mattson and Bruhn (2001) ex-

ment uncertainty and random variability in the

tended the technique to include uncertainty in

landscape?

the initial scarp angle.

Pluvial shoreline scarps form by wave-cut ac-

Figure 2.9 presents an oblique aerial view of

tion during a prolonged period of lake-level high

the shoreline in east-central Tule Valley as it cuts

stand. Following lake-level fall, scarps formed in

across several distinct alluvial fan terraces. This

unconsolidated alluvium evolve to an angle of re-

¬gure illustrates that scarp height is controlled

pose by mass movements and then further evolve

primarily by the slope of the fan terrace, with

by diffusive hillslope processes.

the tallest scarps formed in steeply dipping al-

Historical scarp-analysis methods come in two

luvial fan terraces and debris cones close to the

basic types: midpoint-slope methods and full-

mountain front.

scarp methods. Mid-point slope methods can be

The geometry of a general scarp is de¬ned

further divided into two types. In the ¬rst type,

by the scarp height 2a, initial gradient ±, and

the midpoint slope of a single scarp is inverted

far-¬eld gradient b (Figure 2.10a). The initial and

for diffusion age using the analytic solution

far-¬eld gradients can be combined for the pur-

to the diffusion equation (Andrews and Hanks,

poses of scarp analysis into a single variable: the

1985). In the second approach, the midpoint gra-

reduced initial gradient ± ’ b. The analytic solu-

dient is plotted versus scarp offset for a collection

tion to the diffusion equation for a general scarp

of many scarps with different heights (Bucknam

2.2 ANALYTIC METHODS AND APPLICATIONS 43

House Range

Dome Canyon

Bonneville scarp

lacustrine

deposits

8 m vert.

is mathematically simpler because the derivative

Fig 2.9 Virtual oblique aerial photograph (i.e. aerial photo

draped over digital topography) of pluvial features on the east of Eq. (2.46) is a more compact expression:

side of central Tule Valley. Scarp height is primarily controlled

‚h(x, t) (± ’ b) x + a/(± ’ b)

’b = √

erf

by the slope of the alluvial-fan and debris-cone deposits, i.e.

‚x 2 4κt

wave-cut erosion of steeply dipping deposits tends to

x ’ a/(± ’ b)

produce steep, tall scarps. Inset photo shows 8 m tall scarp at

’ erf √ (2.47)

the entrance to Dome Canyon. 4κt

The left-hand side of Eq. (2.47) is called the re-

duced gradient pro¬le (i.e. the gradient pro¬le

was given by Hanks and Andrews (1989) as

reduced by b, in order to take the far-¬eld gra-

κt x+a/(±’b))2 x’a/(±’b))2 dient into account). Equations (2.46) and (2.47)

e’ ’ e’

h(x, t) = (± ’ b) 4κt 4κt

π are plotted in Figure 2.1 for a range of diffu-

(± ’ b) x + a/(± ’ b) sion ages κt and reduced initial scarp gradients,

a

+ x+ √

erf

±’b ± ’ b. Figure 2.10b illustrates that the midpoint

2 4κt

x ’ a/(± ’ b) of the scarp is insensitive to age for young, tall

a

’ x’ + bx

√

erf

±’b scarps. Note that in Figure 2.9, the units of the

4κt

x-axis are scaled to the scarp width x0 , which, in

(2.46)

turn, is scaled to the scarp height through the re-

lationship x0 = a/(± ’ b). Figure 2.10c illustrates

Full-scarp analyses can be performed by compar-

ing the measured elevation pro¬le of a scarp to that, as the reduced initial gradient decreases,

Eq. (2.46), or by taking the derivative of the ele- the age-diagnostic portion of the scarp becomes

vation pro¬le and working with the slope or gra- progressively concentrated away from the mid-

dient pro¬le. Working with the gradient pro¬le point towards the top and bottom of the scarp.

44 THE DIFFUSION EQUATION

These plots suggest that full-scarp methods are

b (a)

a necessary to extract the independent controls

1

of scarp age and initial gradient on scarp mor-

h0

h phology.

h0 Alternatively, a brute-force approach can be

used which computes analytic or numerical so-

x0

0

lutions corresponding to a full range of model

parameters and then plots the error between

the model and measured pro¬les as a function

1 of each parameter value (e.g. diffusion age and

initial gradient). The diffusion age is then con-

1 strained by the model pro¬le that has the lowest

(b)

error relative to the measured data. This brute-

b force approach is most often used, and it has the

advantage that the goodness-of-¬t can be evalu-

a-b

ated visually on a pro¬le-by-pro¬le basis. Avouac

0.5 (1993) and Arrowsmith et al. (1998) pioneered this

kt = 1.0 technique, assuming initial scarp angles between

30—¦ and 35—¦ . Mattson and Bruhn (2001) improved

= 0.3

kt = 0.01

upon this approach by including a range of initial

= 0.03

= 0.1 gradients in the analysis. Mattson and Bruhn™s er-

0 ror plots are thus two-dimensional, including a

0 1 2 3

range of both diffusion age and reduced initial

angle in the analysis.

0.5 kt = 0.25, 0.5, 1

(c) a ’ b = 0.5 Figures 2.11a and 2.11b illustrate the full-scarp

0.4 method on a short and a tall scarp, respectively,

kt = 0.25, 0.5, 1 each selected from the set of Bonneville scarps.

a - b = 0.35

0.3 The reduced gradient is plotted as a function of

kt = 0.25, 0.5, 1

-b distance along the pro¬le. To the right of each

a ’ b = 0.2

0.2 plot is a grayscale map that illustrates the rel-

ative least-square error between the model and

0.1 observed pro¬les for a range of diffusivity val-

ues and reduced initial gradients. In both maps,

0.0 black colors represent the lowest mismatch and

0 4 6

2

x/x0 X marks the spot of the optimal ¬t. The loca-

tion of this best-¬t solution in the model pa-

Fig 2.10 (a) Analytic solutions of the linear diffusion rameter space simultaneously determines diffu-

equation, starting from an initial gradient ±, scarp height 2h 0 , sivity and reduced initial gradient on a pro¬le-by-

and far-¬eld slope b. The initial scarp width, 2x 0 , is related to

pro¬le basis. In the short-scarp example case, the

height and reduced initial gradient by x 0 = h 0 /(±±b). (b)

best-¬t solution corresponds to κ = 0.95 m2 /kyr

Plots of reduced gradient corresponding to the pro¬les in (a).

and ± ’ b = 0.70. The black band running across

For small κt values (< 0.1), no change in the midpoint slope is

the grayscale map illustrates that, although the

observed, illustrating the insensitivity of the midpoint slope to

best-¬t solution occurs for ± ’ b = 0.70, the solu-

diffusion age for young scarps. (c) Plots of reduced gradient

tion is essentially independent of the reduced ini-

versus distance along pro¬le, for families of solutions

corresponding to a range of morphologic ages (κt = 0.25, tial gradient. This makes physical sense because,

0.5, 1) and a range of reduced initial gradients (± ’ b = 0.2, for a short, narrow scarp, diffusive smoothing

0.35, 0.5). quickly reduces the maximum slope to a value

far below the angle of repose. For a tall, broad

scarp (Figure 2.11b), however, the scarp form is

2.2 ANALYTIC METHODS AND APPLICATIONS 45

of the slope break at the top and bottom of the

0

short scarp (a) scarp. The reduced initial gradient is determined

0.4

by the shape of the ¬‚at, central portion of the gra-

k

0.3 dient pro¬le shown in Figure 2.11b (also shown

’b

in the analytic solutions of Figure 2.1f).

0.2 k = 0.95 m2/kyr

a ’ b = 0.70 7

2.2.8 Degradation of archeological ruins

a’ b 1.4

0.4

0.1

In the summer of 2006 I performed ¬eld work in

Tibet with Jennifer Boerner. Tibet is full of impor-

0

0

tant archeological ruins, and I wondered whether

tall scarp

0.4

the degradation of stone-wall ruins could be de-

k = 0.78 m2/kyr

a ’ b = 0.48

k

scribed by a diffusion model. Stone walls begin

0.3

as consolidated, vertical structures. Over time

’b

scales of centuries to millennia following aban-

0.2

donment, however, mortar will degrade, stones

7

a’ b 1.4

0.4

will topple, and surface processes (creep, biotur-

0.1

bation, eolian deposition, etc.) will act to smooth

(b)

0

the surface.

’30 ’20 ’10 0 10 20 30

x (m) Figure 2.12 illustrates the application of the

diffusion equation to stone ruins in western Ti-

Fig 2.11 Full-scarp curve-¬tting method illustrated for (a)

bet. Figure 2.12a illustrates the solution to the

short and (b) tall example Bonneville scarps. The reduced

diffusion equation with an initially vertical struc-

gradient is plotted as a function of distance from the scarp

midpoint. Grayscale maps to the right of each plot illustrate ture. This equation is identical to the solution for

the relative error between the observed and modeled scarp the gradient pro¬le of a scarp, Eq. (2.47):

as a function of diffusivity κ and reduced initial gradient ± ’ b.

x + x0 x ’ x0

h0

In both maps, black colors represent the lowest mismatch h(x, t) = ’ erf

√ √

erf (2.48)

2x0 4κt 4κt

and X marks the spot of the optimal ¬t. The location of this

best-¬t solution in the model parameter space simultaneously

where h 0 is the initial height of the wall and

determines diffusivity and reduced initial gradient on a

x0 is the half-width. Figure 2.12b presents an

pro¬le-by-pro¬le basis. In the short-scarp example case in (a),

oblique view of one example of stone ruins in

the best-¬t solution corresponds to κ = 0.95 m2 /kyr and

Tibet. These degraded walls are up to 0.5 m tall

± ’ b = 0.70. The black band running across the grayscale

and have spread out to cover a width of several

map illustrates that, although the best-¬t solution occurs for

± ’ b = 0.70, the solution is essentially independent of the meters. Figure 2.12d presents a contour plot of

reduced initial gradient. This makes physical sense because, a detailed topographic survey of a small section

for a short, narrow scarp, diffusive smoothing quickly reduces of one of these ruins. Figure 2.12c plots two to-

the maximum slope to a value far below the angle of repose.

pographic pro¬les extracted from Figure 2.12d.

For a tall, broad scarp (b), however, the solution is sensitive

These plots are compared with Eq. (2.48) for κt

to the initial angle. The best-¬t model parameters in this case

values of 2.0 m2 and 1.3 m2 . Radiocarbon ages in-

have well-de¬ned values for both diffusion age and reduced

dicate a site abandonment age of 1.7 ka at this

initial gradient (κ = 0.78 m2 /kyr and ± ’ b = 0.48).

location. Diffusivity values for these walls, there-

fore, are 1.2 and 0.8 m2 /kyr, respectively. These

values are broadly comparable to the represen-

sensitive to the initial angle. The best-¬t model

tative value for the Basin and Range.

parameters in this case have well-de¬ned values

for both diffusion age and reduced initial gradi-

ent (κ = 0.78 m2 /kyr and ± ’ b = 0.48). Physically, 2.2.9 Radionuclide dispersion in soils

¬tting the full scarp works best because the scarp Thus far we have only considered applications

pro¬le contains independent information about of the diffusion equation to landform evolution.

the diffusion age and the initial gradient. The dif- Many applications of the diffusion equation are

fusion age is determined primarily by the width applicable in other aspects of surface processes,

46 THE DIFFUSION EQUATION

(a) 1 (b)

h(x)

0.5

= 1.0

= 0.3

kt/x02 = 0.01

= 0.03

degraded walls

= 0.1

0

’3 ’2 ’1 0 1 2 3

x/x0

(d) 5cm contours

(c)

0.6

5m

h(x)

kt = 1.3 m2

(m)

kt = 2.0 m2

0.4

0.2

profiles

0

2 4

0 8

6

x (m)

and 95%, suggesting that very little 137 Cs has dif-

Fig 2.12 Diffusion modeling of the stone-wall degradation

fused below 6 cm in these pro¬les. The radionu-

in western Tibet. (a) Solutions to the diffusion equation with

an initially vertical structure. (b) Oblique view of degraded clide pro¬le shape may re¬‚ect the in¬‚uence of

walls in Tibet. (c) Plots of topographic pro¬les perpendicular both surface erosion/deposition and redistribu-

to the wall, with best-¬t solutions corresponding to tion within the soil column. However, in the case

κt = 2.0 m2 and 1.3 m2 . (d) Contour map of a

of Fortymile Wash it is reasonable to assume that

microtopographic survey of degraded walls in western Tibet

surface erosion and deposition were negligible

(near Kyunglung), with pro¬le locations noted.

during the past 50 yr. Except for the active chan-

nel, all of the older surfaces have not experienced

however. Here we consider the migration of signi¬cant ¬‚ooding for several thousand years.

radionuclides into the subsurface within a soil Eolian deposition rates inferred for the Fortymile

pro¬le. Wash fan by Reheis et al. (1995) also suggest that

Beginning in the mid-1950s and continuing eolian erosion/deposition is less than or equal to

through the mid-1960s, above-ground nuclear several millimeters over the 50-yr time scale since

tests introduced radioactive fallout 137 Cs into the the introduction of fallout 137 Cs. Due to the rela-

soil (He and Walling, 1997). Fourteen 137 Cs pro- tive stability of the Fortymile Wash fan surfaces

¬les from the mapped area of Fortymile Wash fan to both ¬‚uvial and eolian erosion/deposition, we

expect the shape of the 137 Cs pro¬le to predom-

were collected and analyzed (Table 2.1) in order to

determine the rate of Cs migration into the soil inantly re¬‚ect redistribution processes. The ex-

over the past 50 years. The Fortymile Wash fan is ception is the active channel. In this case the Cs

of particular interest because it is the depozone pro¬le may be strongly in¬‚uenced by ¬‚uvial mix-

for sediment eroded from the Nevada Test Site. ing in addition to in¬ltration and other mixing

Bulk samples were collected at 0--3 cm, 3--6 cm, processes.

and 6--9 cm. The fraction of total 137 Cs in the up- The equation describing the evolution of ra-

per half of the pro¬le typically varied between 80 dionuclide concentration C by diffusive processes

2.2 ANALYTIC METHODS AND APPLICATIONS 47

«C(z,t)/(C0 dw )

C(z,t)/(C0 dw )

0.2 0.4 0.6 0.8 1

0

0.4

0.2

0 0.6

0

0

2

t = 30 5

t = 20

4

z (cm) sandyD = 6.1D = 5.1

peat,

t = 10 yr

z (cm) loam,

calc. sandy loam, D = 3.5

10 clayey loam, D = 2.0

6

8

cumulative