Modification of wave-cut and faulting-controlled landforms

From a casual observation that the form of degraded fault scarps resembles the error function, this investigation proceeds through an elementary diffusion equation representation of landform evolution to the application of the resulting equations to the modern topography of scarplike landforms. The morphologic observations can be analyzed either in the form of one or more cross-strike elevation profiles or in the form of the slope-offset plot, a point plot of maximum scarp slope versus scarp offset. Working with either or both of these data representations for nine geologic structures, which range in age from 3 to 400 ka B.P. and in offset from 1 to 50 m, we apply analytical solutions for the vertical initial value scarp, the vertical continuous offset scarp, and the finite slope, initial value scarp. The model calculations are intrinsically ambiguous, yielding as the final answer only the product κt (in the case of the initial value problem) or the product κA−1 (in the case of the repeated faulting problem); here t is the age of a single scarp-forming event, 2A is the vertical slip rate, and κ is the “mass diffusivity.” A single profile across three sea cliffs along the Santa Cruz, California, coast is analyzed as three separate initial value problems. A reasonably constrained age for the sea cliff standing above the Highway 1 platform returns κ = 11 GKG (1 GKG = 1 m2/ka). With this κ, we can date the two older sea cliffs. In fact, we do the converse: age estimates for these two older sea cliffs based on a uniform rate of uplift both yield the same κ as for the lower sea cliff. We treat a single profile of the Raymond fault in Pasadena/San Marino in terms of the repeated faulting problem; for it the uplift rate of R. Crook and others yields κ = 16 GKG. The very substantial preexisting offset across the Raymond fault must have been buried/leveled some 230 ka B.P., when the modern topography began to form. Our analysis of the Lake Bonneville shoreline scarps reveals a dependence of κt on 2a, suggestive of nonlinear modification processes. This appearance is treated with the finite slope initial value scarp model to determine κ=1.1 GKG for the Lake Bonneville shoreline scarps. The suggestion of M. N. Machette that approximately 100,000-year-old, meter-high scarps are “unobservable” in weakly consolidated alluvial terranes of the Basin and Range and Rio Grande Rift Valley provinces can be formulated as κ ≳ 1 GKG. The coincidence between this inequality and the Lake Bonneville shoreline κ is striking, and it suggests that the value of κ = 1 GKG may be generally applicable, as a good first approximation, to the modification of alluvial terranes within the semiarid regions of the western United States. The Lake Bonneville shoreline κ is the basis for dating four sets of fault scarps in west-central Utah. The Drum Mountains fault scarps can be modeled in several different circumstances, but the most likely interpretation is that these fault scarps formed as the result of a single episode of normal faulting 3.6 to 5.7 ka B.P. The younger age is associated with quite low initial slope angles (25°). The other three sets of fault scarps show no evidence for finite initial value slopes. Fault scarps along the eastern base of the Fish Springs Range are very young, 3 ka B.P. We estimate the age of fault scarps along the western flank of the Oquirrh Mountains to be 32 ka B.P., which meets the weak geologic constraint that they be older than the Lake Bonneville shoreline. Fault scarps along the northeastern margin of the Sheeprock Mountains are even older, 53 ka B.P. An intriguing consequence of our single-event analysis of these scarps is that an 11.5-m offset occurred in a single earthquake.