Why is topography fractal?

Jon D. Pelletier

Home/Archive/arXiv

arXiv · physics/9705033

Why is topography fractal?

Jon D. Pelletier

View original source ↗Read source document ↗Export metadata ↓

Abstract

The power spectrum S of linear transects of the earth's topography is often observed to be a power-law function of wave number k with exponent close to -2: S(k) is proportional to k^-2. In addition, river networks are fractal trees that satisfy many power-law or fractal relationships between their morphologic components. A model equation for the evolution of the earth's topography by erosional processes which produces fractal topography and fractal river networks is presented and its solutions compared in detail to real topography. The model is the diffusion equation for sediment transport on hillslopes and channels with the local diffusivity proportional to the square of the discharge. The dependence of diffusivity on discharge follows from fundamental equations of sediment transport. We study the model in two ways. In the first analysis the diffusivity is parameterized as a function of relief and a Taylor expansion procedure is carried out to obtain a differential equation for the landform elevation which includes the spatially-variable diffusivity to first order in the elevation. The solution to this equation is a self-affine or fractal surface with linear transects that have power spectra S(k) is proportional to k^-1.8, independent of the age of the topography, consistent with observations. The hypsometry produced by the model is skewed such that lowlands make up a larger area than highlands as in real topography. In the second analysis we include river networks explicitly in a numerical simulation by calculating... Abstract continued in paper.

Explore related subjects

physics.geo-ph cond-mat

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jon D. Pelletier. 1997-05-26. Why is topography fractal?. https://arxiv.org/abs/physics/9705033

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Predicting the Elastic Properties of a Cemented Granular Material during Chemical Damage (Debonding)

While underground reservoirs emerge as essential elements to face global warming, these systems represent complex multi-physical and multiscale problems. The considered injection of fluids during hydrogen storage, carbon dioxide sequestration, or geothermal energy recovery involves a modification of the chemical equilibrium of the fluid in the porous reservoir. Chemical reactions can induce microstructural changes of the rock matrix, leading to a reduction of elastic properties of the material, and to potential settlement or stress redistribution. Consequently, it becomes pivotal to establish predictive behavior laws to describe the effect of chemical damage on elastic properties. Facing the difficulties to estimate experimentally the impact of chemical damage on mechanical properties, a Digital Rock Physics approach is proposed in this contribution. This numerical homogenization scheme is used to compare two distinct types of microstructure models: the first one consists in a Discrete Element Model, while the second one employs a continuous description. This continuous formulation is based on a Phase-Field description to predict the evolution of the microstructure subjected to chemical alterations and on the Fast Fourier Transform to estimate the macroscopic properties of the material. Finally, these frameworks establish different softening laws that can be used as constitutive ingredients for a cemented material during its weathering.

physics.geo-ph↗

arXiv2026

Determination of Physical Height Differences from Time Transfer via the ACES Mission -- A Simulation Study

The determination of physical height differences using highly stable atomic clocks has emerged as a novel approach in relativistic geodesy, exploiting the gravitational redshift as a direct observable of geopotential differences. In this study, we investigate the feasibility of satellite-based clock comparisons using the Atomic Clock Ensemble in Space (ACES) onboard the International Space Station, which enables time transfer via microwave (MWL) and optical (ELT) links. Since operational optical data are not yet available, a comprehensive full-scale simulation of realistic ACES observation scenarios is performed, including detailed noise models of clocks and links. A slope-based estimation method is applied to time series of clock comparisons in order to extract the relativistic redshift signal and derive height differences between the ground stations. The performance of the approach is evaluated for quasi-common view, non-common view, and split non-common view configurations, where the latter divides the observation period into shorter intervals. The results show that optical links enable faster convergence and can achieve height accuracies at the decimeter level within a few days and at the centimeter level over longer periods, while microwave links are more strongly affected by noise and bias contributions. Non-common view processing significantly increases observation availability with only minor loss in accuracy, and the split approach provides robust solutions for larger networks. These findings demonstrate the strong potential of satellite-based clock comparisons as a remote-sensing technique for determining physical height differences on a continental scale.

physics.geo-ph↗

arXiv2026

Sensitivity of neutrino oscillations to the Earth's interior properties

Understanding the Earth s internal structure remains a major challenge, as traditional geophysical methods face ambiguities in linking seismic observations to temperature, composition, or mass density variations. Atmospheric neutrinos offer a complementary probe: while traversing the Earth, they undergo flavor oscillations that depend on the local electron density, which reflects both mass density and composition.

physics.geo-ph↗