Scaling and multifractal fields in the solid earth and topography
Bibliographic record
Abstract
Abstract. Starting about thirty years ago, new ideas in nonlinear dynamics, particularly fractals and scaling, provoked an explosive growth of research both in modeling and in experimentally characterizing geosystems over wide ranges of scale. In this review we focus on scaling advances in solid earth geophysics including the topography. To reduce the review to manageable proportions, we restrict our attention to scaling fields, i.e. to the discussion of intensive quantities such as ore concentrations, rock densities, susceptibilities, and magnetic and gravitational fields. We discuss the growing body of evidence showing that geofields are scaling (have power law dependencies on spatial scale, resolution), over wide ranges of both horizontal and vertical scale. Focusing on the cases where both horizontal and vertical statistics have both been estimated from proximate data, we argue that the exponents are systematically different, reflecting lithospheric stratification which – while very strong at small scales – becomes less and less pronounced at larger and larger scales, but in a scaling manner. We then discuss the necessity for treating the fields as multifractals rather than monofractals, the latter being too restrictive a framework. We discuss the consequences of multifractality for geostatistics, we then discuss cascade processes in which the same dynamical mechanism repeats scale after scale over a range. Using the binomial model first proposed by de Wijs (1951) as an example, we discuss the issues of microcanonical versus canonical conservation, algebraic ("Pareto") versus long tailed (e.g. lognormal) distributions, multifractal universality, conservative and nonconservative multifractal processes, codimension versus dimension formalisms. We compare and contrast different scaling models (fractional Brownian motion, fractional Levy motion, continuous (in scale) cascades), showing that they are all based on fractional integrations of noises built up from singularity basis functions. We show how anisotropic (including stratified) models can be produced simply by replacing the usual distance function by an anisotropic scale function, hence by replacing isotropic singularities by anisotropic ones.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".