Jon D. Pelletier

Scale-invariance of soil moisture variability and its implications for the frequency-size distribution of landslides

Abstract Power spectral analyses of soil moisture variability are carried out from scales of 100 m to 10 km on the microwave remotely-sensed data from the Washita experimental watershed during 1992. The power spectrum S(k) has an approximate power-law dependence on wave number k with the exponent −1.8. This behavior is consistent with the behavior of a stochastic differential equation for soil moisture at a point, and it has important consequences for the frequency-size distribution of landslides. We present the cumulative frequency-size distributions of landslides induced by precipitation in Japan and Bolivia as well as landslides triggered by the 1994 Northridge, California earthquake. Large landslides in these regions, despite being triggered by different mechanisms, have a cumulative frequency-size distribution with a power-law dependence on area with an exponent ranging from −1.5 to −2. We use a soil moisture field with the above statistics in conjunction with a slope stability analysis to model the frequency-size distribution of landslides. In our model, landslides occur when a threshold shear stress dependent on cohesion, pore pressure, internal friction and slope angle is exceeded. This implies a threshold dependence on soil moisture and slope angle since cohesion, pore pressure and internal friction are primarily dependent on soil moisture. The cumulative frequency-size distribution of domains of shear stress greater than a threshold value with soil moisture modeled as above and topography modeled as a Brownian walk is a power-law function of area with an exponent of −1.8 for large landslide areas. This distribution is similar to that observed for landslides. The effect of strong ground motion from earthquakes lowers the shear stress necessary for failure, but does not change the frequency-size distribution of failed areas. This is consistent with observations. This work suggests that remote sensing of soil moisture can be of great importance in monitoring landslide hazards and proposes a specific quantitative model for landslide hazard assessment.

Long-range persistence in climatological and hydrological time series: analysis, modeling and application to drought hazard assessment

We present power spectra of time-series data for tree ring width chronologies, atmospheric temperatures, river discharges and precipitation averaged over hundreds of stations worldwide. The average power spectrum S for each of these phenomena is found to have a power-law dependence on frequency with exponent −12 : S(f) ∝f−12. An advection-diffusion model of the vertical transport of heat and water vapor in the atmosphere is presented as a first-order model of climatic and hydrological variability. The model generates variability with the observed spectrum. The model is validated with a correlation analysis of temperature and water vapor concentration measurements from the TIROS operational vertical sounder (TOVS). Drought frequency analyses based on synthetic lognormal streamflows with the above power spectrum are presented. We show that the presence of long memory as implied by the power-law power spectrum has a significant effect on the likelihood of extended droughts compared with the drought hazard implied from standard autoregressive models with short memory.

Mountains, monsoons, and megafans

In certain cases, the rivers draining mountain ranges create unusually large fan-shaped bodies of sediment that are referred to as fluvial megafans. We combine information from satellite imagery, monthly discharge and precipitation records, digital elevation models, and other sources to show that the formation of fluvial megafans requires particular climatic conditions. Specifically, modern fluvial megafans in actively aggrading basins are produced by rivers that undergo moderate to extreme seasonal fluctuations in discharge that result from highly seasonal precipitation patterns. The global distribution of modern megafans is primarily restricted to 158-358 latitude in the Northern and Southern Hemi- spheres, corresponding to climatic belts that fringe the tropical climatic zone. No rela- tionship exists between megafan occurrence and drainage-basin relief or area. The ten- dency of rivers with large fluctuations in discharge to construct megafans is related to the instability of channels subject to such conditions. Because of the correlation between sea- sonal precipitation and megafan occurrence, the recognition of fluvial megafan deposits in ancient stratigraphic successions may provide critical information for paleoclimate reconstructions.

Wind erosion in the Qaidam basin, central Asia: Implications for tectonics, paleoclimate, and the source of the Loess Plateau

Liquid water and ice are the dominant agents of erosion and sediment transport in most actively growing mountain belts. An exception is in the western Qaidam basin along the northeastern margin of the Tibetan Plateau, where wind and windblown sand have sculpted enormous yardang fields in actively folding sedimentary strata. Here, we present observations suggesting that since the late Pliocene, wind episodically (during glacial and stadial periods) removed strata from the western Qaidam basin at high rates (>0.12–1.1 mm/yr) and may have accelerated rates of tectonic folding. Severe wind erosion likely occurred during glacial and stadial periods when central Asia was drier and the main axis of the polar jet stream was located ~10° closer to the equator (over the Qaidam basin), as predicted by global climate models. Reconstructed wind patterns, the estimated volume of Qaidam basin material removed by wind, and numerical models of dust transport all support the hypothesis that the Qaidam basin was a major source of dust to the Loess Plateau.

Analysis and Modeling of the Natural Variability of Climate

After removing annual variability, power spectral analyses of local atmospheric temperature from hundreds of stations and ice core records have been carried out from timescales of 1 day to 200 kyr. A clear sequence of power-law behaviors is found as follows: 1) from 40 kyr to 200 kyr a flat spectrum is observed, 2) from 2 kyr to 40 kyr the spectrum is proportional to f22 where f is the frequency, and 3) below timescales of 2 kyr the power spectrum is proportional to f21/2. At timescales of less than 1 month the authors observe that the power spectra of continental stations become propotional to f23/2 while maritime stations continue to have power spectra proportional to f21/2 down to timescales of 1 day. To explain these observations, the authors model the variations in the transport of heat in the turbulent atmosphere by including a noise in the heat flux of a flux-gradient parameterization of convective transport, leading to a stochastic diffusion equation for temperature fluctuations. A correlation analysis of temperature fluctuations vertically in the atmosphere from the TIROS operational vertical sounder is carried out to verify the diffusion model and to estimate the coefficient of vertical eddy diffusivity in the atmosphere. The power spectrum of temperature fluctuations at the earth’s surface expected from this model in a two-layer geometry with thermal and eddy diffusion properties appropriate to the atmosphere and ocean and a radiation boundary condition at the top of the atmosphere exhibits the same spectral form as observed in the instrumental and ice core data.

Recent bright gully deposits on Mars: Wet or dry flow?

Bright gully sediments attributed to liquid water flow have been deposited on Mars within the past several years. To test the liquid water flow hypothesis, we constructed a high-resolution (1 m/pixel) photogrammetric digital elevation model of a crater in the Centauri Montes region, where a bright gully deposit formed between 2001 and 2005. We conducted one-dimensional (1-D) and 2-D numerical flow modeling to test whether the deposit morphology is most consistent with liquid water or dry granular flow. Liquid water flow models that incorporate freezing can match the runout distance of the flow for certain freezing rates but fail to reconstruct the distributary lobe morphology of the distal end of the deposit. Dry granular flow models can match both the observed runout distance and the distal morphology. Wet debris flows with high sediment concentrations are also consistent with the observed morphology because their rheologies are often similar to that of dry granular flows. As such, the presence of liquid water in this flow event cannot be ruled out, but the available evidence is consistent with dry landsliding.

Self-Affine Time Series: II. Applications and Models

Publisher Summary This chapter focuses on the applications and models of self-affine time series. The chapter presents the stochastic component of time series associated with complex phenomena that can have considerable order and, in particular, exhibit self-affine behavior. One of the applications considered is the natural variability of climate. On short time scales, atmospheric temperature observations exhibit deterministic daily and yearly periodicities. The studies of the stochastic variability of climate are important in a variety of ways. They provide an important test of the validity of global circulation models (GCMs). The second application considered is porosity variations in sedimentary basins. A model developed for the growth of atomic surface layers is modified so that it is applicable to the spatial and temporal variations in deposition and erosion. The third application considered is the variability of the earths magnetic field. The field exhibits a binormal behavior and when a fluctuation crosses the zero intensity value a reversal occurs.

Drainage basin evolution in the Rainfall Erosion Facility: dependence on initial conditions

Four experiments in alluvial drainage basin evolution were carried out in the Rainfall Erosion Facility (REF) at Colorado State University to investigate the dependence of basin evolution on initial topography. Basins were initially undissected. Each experiment began with a unique initial condition representing various end-members of relief and hypsometry. Drainage network development, hillslope processes, basin denudation, and basin response to base-level lowering all depended strongly on the initial topography. No classic model of drainage network evolution was found to be generally applicable. Initially, planar slopes first developed subparallel channels that extended headward dendritically during an early phase of extension. Channel incision occurred first in the interior of the basin where saturation overland flow was greatest, not at the basin outlet as assumed in most classic models of network development. Channels widened over time, initiating lateral migration and drainage capture in the downslope portion of the watershed before transferring lateral migration upslope. Planar basins of larger initial gradient grew headward more quickly and become more deeply entrenched, inhibiting late-stage lateral migration. An experiment with initial relief concentrated at a plateau edge evolved in several unique ways. A high ratio of subsurface-to-surface flow gave rise to mass movements at the plateau edge and outlet channels. Deep channels were quickly cut initially but did not extend far upslope because slope instability undermined channel head migration, leaving the plateau undissected and hence very slow to erode. These results suggest that the distribution of relief within a basin exerts an important control on drainage network pattern and basin denudation. In addition, erosional basins may evolve in several distinct modes characterized by particular combinations of hypsometry, hillslope processes, and mean denudation rate.

The power spectral density of atmospheric temperature from time scales of 10−2 to 106 yr

After removing annual variability, power spectral analyses of local atmospheric temperature from hundreds of stations and ice core records have been carried out from time scales of 10−2 to 106 yr. A clear sequence of power-law behaviors is found as follows: (1) from 40 ka to 1 Ma a flat spectrum is observed; (2) from 2 ka to 40 ka the spectrum is proportional to f−2 where f is the frequency; and (3) below time scales of 2 ka the power spectrum is proportional to f−1/2. At time scales less than 1 month we observe that the power spectra of continental stations become proportional to f−3/2 while maritime stations continue to have power spectra proportional to f−1/2 down to time scales of 1 day. To explain these observations, we model the vertical transport of heat in the atmosphere as a stochastic diffusion process. The power spectrum of temperature fluctuations at the earths surface expected from this model equation in a two-layer geometry with thermal and eddy diffusion properties appropriate to the atmosphere and the ocean and a radiation condition at the top of the atmosphere agrees with the observed spectrum. The difference in power spectra between continental and marine stations can be understood with this approach as a consequence of the air mass above a maritime station exchanging heat with both the atmosphere above and the ocean below while a continental station exchanges heat mostly with the atmosphere above.

Quantifying the climatic and tectonic controls on hillslope steepness and erosion rate

Hillslopes in humid regions are typically convex to concave in profile and have a relatively thick, continuous regolith cover. Conversely, hillslopes in arid regions are typically cliff-dominated and have a relatively thin, discontinuous regolith cover. The difference between these two end-member slope forms is classically attributed to climate, but climate, tectonics, and lithology all play a role. In this paper, we describe a mathematical model for hillslope gradient and regolith thickness using basic climatic and tectonic input data for a given rock type. The model first solves for the regolith thickness on a planar slope segment in topographic steady state using the soil production function and a prescribed uplift/incision rate. The climatic and lithologic controls on soil production rates are quantified using an empirical energy-based model for the physical weathering of bedrock. The slope gradient is then computed by balancing uplift/incision rates with sediment fluxes calculated using a nonlinear depth- and slope-dependent sediment transport model. The model quantifies the ways in which, as aridity and uplift/incision rates increase, regolith thicknesses decrease and hillslope gradients increase nonlinearly until a threshold condition is reached, beyond which bare, cliff-dominated slopes form. The model can also be used to estimate long-term erosion rates and soil residence times using basic input data for climate and regolith thickness. Model predictions for erosion rates closely match cosmogenically derived erosion rates in granitic landscapes. This approach provides a better quantitative understanding of the climatic and tectonic controls on slope form, and it provides a simple, widely applicable method for estimating long-term erosion rates and the thickness of regolith cover on hillslopes.