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Entropy Analysis of Multiple Scale Causality and Qualitative Causal Shifts in Spatial Systems
Abstract
Spatial systems are typically characterized by multiple controlling factors and processes operating at different spatial and temporal scales (multiple scale causality [MSC]). An entropy decomposition-based approach to MSC is presented here in two contexts. First, given maps or distributions of an observed phenomenon at two or more scales, the contribution at more local or global (relative to the primary scale of observation) controls to the observed entropy can be estimated. Second, a theoretical treatment of the entropy decomposition equations shows that as the range of scale is increased by broadening or narrowing resolutions or by incorporating more controls, the influence of larger or smaller-scale influences not only changes, but may change qualitatively, e.g., in terms of having positive (entropy-increasing) or negative (information-increasing) effects. Such qualitative causal shifts have implications for efforts to use any single causal explanation across the molecular to planetary spatial and instantaneous to geological range of scales relevant to physical geography. The entropy decomposition method is illustrated with an application to soil landscapes in the Ouachita Mountains, Arkansas.*Dan Marion of the U.S. Forest Service, and other personnel of the USDA Forest Service, Southern Research Station, and Ouachita National Forest assisted in innumerable aspects of the Ouachita Mountains soil research. Linda Martin and Zach Musselman provided comments on an earlier draft of this paper. Two anonymous reviewers made insightful comments that improved the paper substantially. Mistakes and wild-eyed, arm-waving speculations are not their fault.
... The Global land 1 km base elevation project (GLOBE) 1 km Variable by location Macromegarelief multiple scale causality (see Phillips (2005) for more details)). Future advancement of scale dependence will likely require further advancements in data structures that account for multiscale representation and analyses. ...
... This information can be derived at a variety of scales from the original data to provide information on important concepts in geomorphology. Examples include: (1) local and nonlocal variations in sediment flux (Gabet, 2003;Tucker and Bradley, 2010); (2) connectivity within the landscape (Brierley et al., 2006;Bracken and Croke, 2007); and (3) multiple-scale causality (Phillips, 2005). ...
Geomorphologists require quantitative information about the land surface. New sensors can now measure elevation changes at a variety of scales and these data are used to generate digital terrain model (DTM) that accurately characterize topography. A variety of DTM analytics are used to support a multitude of geomorphological studies. However, there are numerous issues involving representation, sampling, interpolation, and error assessment and correction, which must be addressed before using the elevation data. Data reduction, filtering, and accuracy are key aspects to consider. Knowledge of these issues is critical for terrain analysis and the communication of information derived from a DTM.
... Multilevel models are further differentiated by their focus on either a univariate statistic or a multiple regression context. The former focuses on decomposing a single variable or measure as a function of itself, such as spatial variability (Oliver and Webster 1986;Collins and Woodcock 2000), moving window averages (Pigozzi 2004), the statistical likelihood (Kolaczyk and Huang 2001), diversity and dissimilarity indices (Wong 2003;Manley et al. 2019), or entropy (Phillips 2005;Batty 2010;Leibovici and Birkin 2015). In contrast, the latter focuses on decomposing a variable as a function of other variables (e.g., Duncan and Jones 2000). ...
Scale is a central concept in the geographical sciences and is an intrinsic property of many spatial systems. It also serves as an essential thread in the fabric of many other physical and social sciences, which has contributed to the use of different terminology for similar manifestations of what we refer to as ‘scale’, leading to a surprising amount of diversity around this fundamental concept and its various ‘multiscale’ extensions. To address this, we review common abstractions about spatial scale and how they are employed in quantitative research. We also explore areas where the conceptualizations of multiple spatial scales can be differentiated. This is achieved by first bridging terminology and concepts, and then conducting a scoping review of the topic. A typology for spatial scale is discussed that can be used to categorize its multifarious meanings and measures. This typology is then used to distinguish what we term ‘process scale,’ from other types of spatial scale and to highlight current trends in uncovering aspects of process scale. We end with suggestions on how to further build knowledge regarding spatial processes through the lens of spatial scale.
... The arguments that follow are taken from Phillips (2005), who took a starting point from physics and generalized to the more complex systems of geography. ...
Geography and geosciences deal with phenomena that span spatial scales from the molecular to the planetary, and temporal scales from instantaneous to billions of years. A strong reductionist tradition in geosciences and spatial sciences tempts us to seek to apply similar representations and process-based explanations across these vast-scale ranges, usually from a bottom-up perspective. However, the law of scale independence (LSI) states that for any phenomenon that exists across a sufficiently large range of scales, there exists a scale separation distance at which the scales are independent with respect to system dynamics and explanation. The LSI is evaluated here from five independent perspectives: geographic intuition, dynamical systems theory, Kolmogorov entropy, hierarchy theory, and algebraic graph theory. All of these support the LSI. Results indicate that rather than attempting to identify the largest or smallest relevant scales and work down or up from there, the LSI dictates a strategy of focusing directly on the most important or interesting scales. An example is given from a hierarchical state factor model of ecosystem responses to climate change. ARTICLE HISTORY
... The first and most important application of S in landscape ecology may be the establishment of a linkage between landscape patterns/dynamics and ''the central organizing theories or principles of nature'' (Cushman 2015, p. 8). In addition, S can be used as an alternative to Shannon entropy in the entropic analysis of landscapes, as summarized by Vranken et al. (2015), to quantify landscape heterogeneity (e.g., Díaz-Varela et al. 2016;Huang et al. 2016), to describe the instability of landscape evolution (e.g., Newman 1999;Zaccarelli et al. 2013), and to study the effect of scale on landscape patterns (e.g., O'Neill et al. 1989;Johnson et al. 2001;Phillips 2005). ...
Context
The second law of thermodynamics is a central organizing principle of nature, whose core concept, Boltzmann entropy, is fundamentally important in landscape ecology research. However, the use of this entropy has remained at a conceptual level in landscape ecology for one and a half centuries. It was not until very recently that methods were developed for computing the Boltzmann entropy of landscape gradients and mosaics.
Objectives
The aim of this study was to examine the computational method (i.e., resampling-based method) for landscape gradients. The first objective was to validate whether the Boltzmann entropy computed using this method was thermodynamically consistent (i.e., consistent with statistical thermodynamics). The second objective was to propose a different method for computing thermodynamically consistent entropy.
Methods
A kinetic-theory-based approach was proposed for testing the thermodynamic consistency of entropy. This approach was applied to both relative and absolute Boltzmann entropies by the resampling-based method, revealing that the absolute Boltzmann entropy is only partly consistent. Hypothesis-driven experiments were designed to determine the cause.
Results
The cause was demonstrated to be the generalization technique for generating the macrostate of a landscape gradient, which is called resampling. A different computational method was developed on the basis of an alternative technique (i.e., aggregation).
Conclusions
Validation of its thermodynamic consistency is necessary even if a “thermodynamic” entropy is computed strictly according to the formula. The entropy computed using the aggregation-based method passed the validation and is recommended to be used in linking landscape ecological processes with statistical thermodynamics.
... Such problems crop up, and remain challenging, with respect to issues such as multiple-resolution models, upscaling, and downscaling. However, the rules are typically not constant across scales, which is consistent with intuition, empirical evidence, and dynamical systems theory (Phillips, 1986(Phillips, , 1988(Phillips, , 2005. Can, for instance, the global biogeography of ants shed light on the biogeomorphic impacts of spatial foraging strategies or nest site selections of particular ant species (or vice-versa)? ...
Scale linkage problems in geosciences are often associated with a hierarchy of components. Both dynamical systems perspectives and intuition suggest that processes or relationships operating at fundamentally different scales are independent with respect to influences on system dynamics. But how far apart is “fundamentally different”—that is, what is the “vanishing point” at which scales are no longer interdependent? And how do we reconcile that with the idea (again, supported by both theory and intuition) that we can work our way along scale hierarchies from microscale to planetary (and vice-versa)? Graph and network theory are employed here to address these questions. Analysis of two archetypal hierarchical networks shows low algebraic connectivity, indicating low levels of inferential synchronization. This explains the apparent paradox between scale independence and hierarchical linkages. Incorporating more hierarchical levels results in an increase in complexity or entropy of the network as a whole, but at a nonlinear rate. Complexity increases as a power α of the number of levels in the hierarchy, with α and usually ≤ 0.6. However, algebraic connectivity decreases at a more rapid rate. Thus, the ability to infer one part of the hierarchical network from other level decays rapidly as more levels are added. Relatedness among system components decreases with differences in scale or resolution, analogous to distance decay in the spatial domain. These findings suggest a strategy of identifying and focusing on the most important or interesting scale levels, rather than attempting to identify the smallest or largest scale levels and work top-down or bottom-up from there. Examples are given from soil geomorphology and karst flow networks.
... Recent papers highlight the continued importance of scale in geomorphology (e.g. Livingstone, 1995, 2001; Lane and Richards, 1997; Lane, 2001; Viles, 2001; Phillips, 2001 Phillips, , 2005 Cammeraat, 2002; Inkpen, 2004). These papers are an overt expression of the problem of how to deal with scale in any attempt to understand the physical environment as a totality. ...
Absolute views of space and time tend to involve entities and processes as being mapped onto a fixed framework. In contrast, in relational views of space and time, processes fundamentally define the framework and hence define scale. A simple indicator or representation of erosion, point height changes, provides an example of how relational views of space and time could be used to analyse geomorphological questions. Defining the behaviour of points in terms of their relation to other points on the surface moves study away from the dichotomy of “inherent” versus “external” explanations of change and instead focuses on the development of patterns of response of the erosional surface. Even this simple weathering data set of changes in point heights reveals common modes of behaviour between points and also allows the emergence of appropriate scales of analysis and explanation.
... Operational problems involve identifying characteristic scales, determining conditions of scale independence, decisions as to what phenomena to represent at a given scale, and tools for transferring information and representations between scales. Theoretical issues of scale linkage involve fundamental questions of whether any single rules, relationships or representations are even potentially valid across all relevant scale ranges; bottom-up vs. top-down inuences, and multiple scale causality (see, e.g., Phillips, 2005a). Geomorphic systems except at the most atomistic level consist of coupled subsystems. ...
Geomorphic systems consist of coupled subsystems with traits of small-world networks (SWN), characterized by tightly connected clusters of components, with fewer connections between the clusters. Geomorphic systems based on scale hierarchies often exhibit a connected caveman small-world network (CCSWN) structure. SWNs are efficient for linking a large number of components with a relatively small number of links; but effects of CCSWN structure on synchronization and scale linkage have not been examined. Synchronization is analyzed via graph theory and applied to: (1) relationships among three levels of form–process interaction in stream channels; (2) hierarchical relationships of weathering systems at scales from weathering profiles to landscapes; and (3) interactions in fluviokarst systems at the scale of flow processes and of landscape evolution. Relationships among system components are represented as simple unweighted graphs. The largest eigenvalue of the adjacency matrix (spectral radius) reflects the critical coupling strength required to synchronize the system. The second-smallest eigenvalue of the Laplacian of the adjacency matrix (algebraic connectivity) is a measure of the synchronizability. In all examples both are much less than the maximum for networks of the same number of nodes. The sparseness of the networks is the major contributor to the low synchronization, but the specific pattern of connections (“wiring”) is also significant. Where CCSWN structures arise naturally, they help explain how geomorphic effects are transmitted between disparate scales in the absence of obvious scale linkage. Where CCSWNs are an option for representation of geomorphic systems in models and data structures, they will not improve scale linkage, despite the efficiency of SWNs in other respects. Methods developed here can be applied to evaluating alternative spatial data structures or mapping strategies which either increase synchronization, supporting a lumping or aggregation approach, or decrease synchronization, indicating disaggregation or splitting into scale hierarchies.
... This type of information is not available from the point data. TMEM data may be, as noted by Inkpen et al. (2004), a good representation of average weathering across sites; but within sites, the change in the scale of focus may be sufficient to produce a qualitative causal shift such as outlined by Phillips (2005). At this more local scale, the same general variables may be of significance, but their mediation at this scale is not capable of the same generalization as the aggregate data. ...
Sites with five periods of consecutive measurements from the Kaikoura Peninsula data were used to assess the influence of erosional history and surface topography upon subsequent erosion rates. Analysis of multilevel models suggests that initial erosion rates and initial surface topography have a significant, but decreasing influence upon subsequent erosion rates. Surface topography and erosion rates immediately prior to each measurement period have a consistent positive and negative relationship respectively to erosion rates in the subsequent measurement periods. These data suggest that there is a short-term memory influencing erosion rates at any specific measurement period. Use of traversing microerosion meter data may, however, not be the most appropriate means of analysing the causes of the variance of erosion at the intra-site scale.
... effects on river regime or migratory organisms) (Allan, 1995). New methods for data analysis may need to be devised or introduced to eco-hydromorphology to: (i) reveal cross-scale pattern and process relationships; and (ii) facilitate scaling up of results from the small spatio-temporal scales often amenable to research, to larger scales relevant to management (Phillips, 2005). ...
The assessment of links between ecology and physical habitat has become a major issue in river research and management. Key drivers include concerns about the conservation implications of human modifications (e.g. abstraction, climate change) and the explicit need to understand the ecological importance of hydromorphology as prescribed by the EU's Water Framework Directive. Efforts are focusing on the need to develop ‘eco‐hydromorphology’ at the interface between ecology, hydrology and fluvial geomorphology. Here, the scope of this emerging field is defined, some research and development issues are suggested, and a path for development is sketched out.
In the short term, major research priorities are to use existing literature or data better to identify patterns among organisms, ecological functions and river hydromorphological character. Another early priority is to identify model systems or organisms to act as research foci. In the medium term, the investigation of pattern–processes linkages, spatial structuring, scaling relationships and system dynamics will advance mechanistic understanding. The effects of climate change, abstraction and river regulation, eco‐hydromorphic resistance/resilience, and responses to environmental disturbances are likely to be management priorities. Large‐scale catchment projects, in both rural and urban locations, should be promoted to concentrate collaborative efforts, to attract financial support and to raise the profile of eco‐hydromorphology.
Eco‐hydromorphological expertise is currently fragmented across the main contributory disciplines (ecology, hydrology, geomorphology, flood risk management, civil engineering), potentially restricting research and development. This is paradoxical given the shared vision across these fields for effective river management based on good science with social impact. A range of approaches is advocated to build sufficient, integrated capacity that will deliver science of real management value over the coming decades.
Copyright © 2007 John Wiley & Sons, Ltd.
... The concept of entropy has long been used in various geographical research (Medvedkov 1967, Marchand 1972, Batty 1976). Recent efforts of using entropy in spatial analysis include, for example, measuring spatial information in maps (Li and Huang 2002), comparing categorical maps (Remmel and Csillag 2006), and examining multi-scale causality in spatial systems (Phillips 2005). Most existing entropy-based approaches focus on (1) deriving a global measure of spatial information and/or (2) analyzing categorical data (e.g., soil types, map symbols, or classified data). ...
The relationship between two or more variables may change over the geographic space. The change can be in parameter values (e.g., regression coefficients) or even in relation forms (e.g., linear, quadratic, or exponential). Existing local spatial analysis methods often assume a relationship form (e.g., a linear regression model) for all regions and focus only on the change in parameter values. Therefore, they may not be able to discover local relationships of different forms simultaneously. This research proposes a nonparametric approach, a local entropy map, which does not assume a prior relationship form and can detect the existence of multivariate relationships regardless of their forms. The local entropy map calculates an approximation of the Rényi entropy for the multivariate data in each local region (in the geographic space). Each local entropy value is then converted to a p-value by comparing to a distribution of permutation entropy values for the same region. All p-values (one for each local region) are processed by several statistical tests to control the multiple-testing problem. Finally, the testing results are mapped and allow analysts to locate and interactively examine significant local relationships. The method is evaluated with a series of synthetic data sets and a real data set.
... x x In the experimental sciences, observed processes and patterns can often be conceptualized as macro-scale manifestations of micro-scale processes. However, in many cases, a more typical situation involves observed patterns or system states that are created or influenced by multiple processes and controls [49]. ...
Cancer is one of the most complex dynamic human disease. Despite rapid advances in the fields of molecular and cell biology, it is still widely debated as to how neoplastic cells progress through carcinogenesis and acquire their metastatic ability. The need to find a new way of observing anatomical entities and their underlying processes, and measuring the changes they undergo, prompted us to investigate the Theory of Complexity, and to apply its principles to human cancer. Viewing a neoplasm as a system that is complex in time and space it is likely to reveal more about its behavioral characteristics, and this manner of thinking may help to clarify concepts, interpret experimental data, indicate specific experiments and categorize the rich body of knowledge on the basis of the similarities and/or shared behaviors of very different tumors.
... In the experimental sciences, observed patterns can often be conceptualized as macro-scale manifestations of microscale processes. However, in many cases, a more typical situation involves observed patterns or system states that are created or influenced by multiple processes and controls [30]. Cancer is determined by a number of processes and controls operating over much broader scales, and by factors such as structural controls that may operate at scales ranging from molecular to environmental [2]. ...
Background
Cancer remains one of the most complex diseases affecting humans and, despite the impressive advances that have been made in molecular and cell biology, how cancer cells progress through carcinogenesis and acquire their metastatic ability is still widely debated.
Conclusion
There is no doubt that human carcinogenesis is a dynamic process that depends on a large number of variables and is regulated at multiple spatial and temporal scales. Viewing cancer as a system that is dynamically complex in time and space will, however, probably reveal more about its underlying behavioural characteristics. It is encouraging that mathematicians, biologists and clinicians continue to contribute together towards a common quantitative understanding of cancer complexity. This way of thinking may further help to clarify concepts, interpret new and old experimental data, indicate alternative experiments and categorize the acquired knowledge on the basis of the similarities and/or shared behaviours of very different tumours.
Landscapes are influenced by processes operating at scales from molecules to planets, and over time spans ranging from instantaneous to billions of years. Thus scale contingency is an innate, unavoidable aspect of landscape evolution, and is common in landscapes. The laws, place, and history factors relevant to landscape evolution may vary with spatial and temporal scale. Scale contingency is epistemological or ontological and leads to issues with scale linkage—transferring relationships and representations across a range of scales. Space/time ratios are useful in determining the range of scales over which landscape phenomena may be (in)dependent, and thus as practical tools in studying landscape evolution. Several different arguments or analyses show that processes or relationships that operate over distinctly different (by at least two orders of magnitude) scales are independent in terms of their effects on system-level dynamics of landscapes.
The paradigm and models of traditional soil science lack the ability to adequately address issues of soil dynamics, environmental integration, and change. Unexplainable research results obtained from traditional soil studies applied to non-traditional soil phenomena in physical geography, archaeology and ecology speak to the current need for soil science to move beyond description and classification and into a dynamic process-oriented soil science capable of providing explanations. Soils do not behave as static inert geologic detritus affected by climate, organisms, relief, and parent material through time, but instead soils behave as self-organizing systems dynamically interrelating with their environment. Recognition of this dynamic behaviour required a re-examination of how scientists in general think and in how modern soil science specifically evolved its basic paradigms and models. This book examines the dynamics of soil organic carbon and demonstrates the self-organizing nature of soil through time as soil responds to a wide range of environmental and human perturbations. Makes soil science accessible to a wider audience by integrating soil science with biology, geography and archaeology Demonstrates universal application by including case studies from around the world Avoids pitfalls of determinism and vitalism by being well founded in the philosophy of science.
Cancer research has undergone radical changes in the past few years. Amount ofinformation both at the basic and clinical levels is no longer the issue. Rather, how tohandle this information has become the major obstacle to progress. System biology is thelatest fashion in cancer biology, driven by advances in technology that have provided uswith a suite of "omics" techniques. It can be seen as a conceptual approach to biologicalresearch that combines "reductionist" (parts) and "integrationist" (interactions) research,to understand the nature and maintenance of entities. In geometrical terms, cancerouslesions can be depicted as fractal entities mainly characterized by their irregular shape,self-similar structure, scaling relationship and non-integer or fractal dimension. It isindubitable that The Fractal Geometry of Nature has provided an innovative paradigm, anovel epistemological approach for interpreting the anatomical world. It is also knownthat mathematical methods and their derivatives have proved to be possible and practicalin oncology. Viewing cancer as a system that is dynamically complex in time and spacewill probably reveal more about its underlying behavioral characteristics. It isencouraging that mathematicians, biologists and clinicians contribute together towards acommon quantitative understanding of cancer complexity.
Based on research of the river landscape, spatial variability of environmental values and their changes over time, it is possible to identify parameters which are in some way involved in the formation of runoff processes. In this paper the issue of the natural values disturbances in an environment of watercourse and its floodplain is solved on the example of two model sites. These are the lowland catchment areas of small watercourses in the Czech Republic. On basis of the map visualization of collected data, the authors deal with interpretation of the landscape spatial pattern within the context of its ecological stability. To analyze the areas, series of maps are used in the appropriate scale showing the selected indicators related to the environmental values. These indicators are a combination of the field research results and information gathered from the available data sources. The space is devoted to the analysis of relationships between selected environmental indicators of spatial character and their impact on the state of the river network. Analyzed are the consequences of human activities in the riparian zone of watercourses, with regard to the phenomenon of the river continuum degradation and changes in other hydrological variables that may affect the overall security of the floodplain area.
Despite years of intensive investigation that has been made in understanding women's susceptibility to breast cancer, it remains a major cause of death worldwide. In mathematical terms, breast cancer can be outlined as a non-linear system that advances in time and in space through different states and a number of transitions from one state to another over a certain time interval. Breast cancer emerges from multiple spontaneous and/or inherited alterations that induce changes in expression patterns of genes and proteins that function in complex networks controlling critical cellular events. Here, we discuss breast cancer as a dynamical disease using the basic principles of the mathematics of non-linear systems and chaos theory. Additionally, some of the critical concepts necessary to give meaning to its underlying physical complexity are introduced. This way of thinking may help to clarify concepts, indicate alternative experiments and categorie the actual knowledge.
Focusing primarily on the actor – human or nonhuman, individual or group, conscious or unconscious – Actor-Network Theory (ANT) explores the interconnectedness of all things. ANT recognises that all objects and things exhibit consciousness, and through a consciousness, interact heterogeneously in space; the location of the interaction(s), where they are performed homogeneously, is the landscape. If, as ANT promotes, all objects and things exhibit consciousness, then the closer in space they are to one another, the more essential they are to each other. These notions have specific ties to the on-going critique in landscape studies of focusing on rural and local scales, and the continuing debate in human (and physical) geography regarding the necessity of scale itself. Using ANT as a framework, and punctuated with true-life anecdotes, this article wraps the landscape-created-by-nature and the landscape-created-by-human debates into a non-dialectic whole, demonstrating, perhaps provocatively and controversially, that any landscape should be distinctly anti-dialectic and removed from scalar constraints. While ANT remains an often-overlooked and misunderstood technique for studying the landscape, this article's crux rests in nothing less than attempting to lay the groundwork for using ANT as a practical technique when studying any landscape in any location at any time (and in any related disciplinary field). If indeed, as argued in this paper and as ANT suggests, everything is networked, then scale, specifically when applied to any landscape, becomes irrelevant.
To provide the accurate alternative metrical means of monitoring the effects of new antiviral drugs on the reversal of newly formed collagen.
Digitized histological biopsy sections taken from 209 patients with chronic C virus hepatitis with different grade of fibrosis or cirrhosis, were measured by means of a new, rapid, user-friendly, fully computer-aided method based on the international system meter rectified using fractal principles.
The following were described: geometric perimeter, area and wrinkledness of fibrosis; the collation of the Knodell, Sheuer, Ishak and METAVIR scores with fractal-rectified metric measurements; the meaning of the physical composition of fibrosis in relation to the magnitude of collagen islets; the intra- and inter-biopsy sample variability of these parameters; the"staging" of biopsy sections indicating the pathway covered by fibrosis formation towards its maximum known value; the quantitative liver tissue architectural changes with the Hurst exponent.
Our model provides the first metrical evaluations of the geometric properties of fibrosis and the quantitative architectural changes of the liver tissue. The representativeness of histological sections of the whole liver is also discussed in the light of the results obtained with the Hurst coefficient.
In geometrical terms, tumour vascularity is an exemplary anatomical system that irregularly fills a three-dimensional Euclidean space. This physical characteristic and the highly variable shapes of the vessels lead to considerable spatial and temporal heterogeneity in the delivery of oxygen, nutrients and drugs, and the removal of metabolites. Although these biological characteristics are well known, quantitative analyses of newly formed vessels in two-dimensional histological sections still fail to view their architecture as a non-Euclidean geometrical entity, thus leading to errors in visual interpretation and discordant results from different laboratories concerning the same tumour. We here review the literature concerning microvessel density estimates (a Euclidean-based approach quantifying vascularity in normal and neoplastic pituitary tissues) and compare the results. We also discuss the limitations of Euclidean quantitative analyses of vascularity and the helpfulness of a fractal geometry-based approach as a better means of quantifying normal and neoplastic pituitary microvasculature.
This book provides a theory to overcome the problem of identifying the principles behind the interdependence of different aspects of nature. Climate, vegetation, geology, landforms, soils, hydrology, and other environmental factors are all linked. Many scientists agree that there must be some general principles about the way in which earth surface systems operate, and about the ways in which the interactions of the biosphere, lithosphere, hydrosphere, and atmosphere manifest themselves. Yet there may be inherent limits on our ability to understand and isolate these interactions using traditional reductionist science. The argument of this book is that the simultaneous presence of order and chaos reflects fundamental, common properties of earth surface processes and systems. It shows how and why this is the case, with examples ranging from evolutionary and geological times scales to microscale examinations of process mechanics.
A pair of models of fractal recursion on city hierarchy, f(m) = f(1)r(f)(1-m) and P-m = P(1)r(p)(m-1) are derived using entropy-maximizing methods, and the relationship of inverse proportion between the number (f(m)) of cities at a given level of the urban hierarchy and the average population size (P-m) of the f(m) cities is established, i.e f(m) proportional to 1/P. It is demonstrated that the underlying rationale of both the scale law of city rank-size distribution and the Zipf dimension value in standard state ( dz = lnr(p)/lnr(f) = 1) rests with maximization of information entropy of city hierarchies.
The historical development of our pedogenetic model is reviewed, and trends or directions toward the future are discussed. Pedogenetic models are characterized with respect to relative degree of computation, complexity, and level of organization. These three characteristics are used as a framework for classification. Early qualitative models have well served the purpose of soil survey to describe the distribution of soils in landscapes. Further development of qualitative models will contribute to the understanding of soil genesis in areas where soil surveys are not completed. In the developed countries, however, the scientific questions have changed dramatically. Greater interest lies in understanding the physical and chemical processes of soil formation acting within relatively short time frames and assessing the interactions between natural processes and environmental change or anthropogenic impacts. Quantification of pedogenetic processes is a life-line to other environmental disciplines, and a mechanistic understanding would, in an ideal situation, describe and predict the behavior of a system for a limited number of years under changing environmental conditions.A proposed approach to mathematical simulation of the dynamic pedogenetic processes is to integrate soil physical, chemical, and other ecological system models into a quantitative pedogenetic model. Models may be developed as research tools in data-intensive studies at the pedon or horizon level, or for the purpose of simulating a soil system at the catena or soil region level. Specific kinds of pedogenetic models have specific implications with respect to the availability and spatial variability of the data needed for development and testing.
Theories of scaling apply wherever similarity exists across many scales. This similarity may be found in geometry and in dynamical processes. Universality arises when the qualitative character of a system is sufficient to quantitatively predict its essential features, such as the exponents that characterize scaling laws. Within geomorphology, two areas where the concepts of scaling and universality have found application are the geometry of river networks and the statistical structure of topography. We begin this review with a pedagogical presentation of scaling and universality. We then describe recent progress made in applying these ideas to networks and topography. This overview leads to a synthesis that attempts a classification of surface and network properties based on generic mechanisms and geometric constraints. We also briefly review how scaling and universality have been applied to related problems in sedimentology—specifically, the origin of stromatolites and the relation of the statistical pro...
This article is written as a prolegomena, both to a research program, and a forthcoming book discussing the same issues in greater detail (Kauffman, 1991). The suspicion that evolutionary theory needs broadening is widespread. To accomplish this, however, will not be easy. The new framework I shall discuss here grows out of the realization that complex systems of many kinds exhibit high spontaneous order. This implies that such order is available to evolution and selective forces for further molding. But it also implies, quite profoundly, that the spontaneous order in such systems may enable, guide and limit selection. Therefore, the spontaneous order in complex systems implies that selection may not be the sole source of order in organisms, and that we must invent a new theory of evolution which encompasses the marriage of selection and self-organization.
Remotely sensed data have been used extensively for environmental monitoring and modeling at a number of spatial scales; however, a limited range of satellite imaging systems often constrained the scales of these analyses. A wider variety of data sets is now available, allowing image data to be selected to match the scale of environmental structure(s) or process(es) being examined. A framework is presented for use by environmental scientists and managers, enabling their spatial data collection needs to be linked to a suitable form of remotely sensed data. A six-step approach is used, combining image spatial analysis and scaling tools, within the context of hierarchy theory. The main steps involved are: (1) identification of information requirements for the monitoring or management problem; (2) development of ideal image dimensions (scene model), (3) exploratory analysis of existing remotely sensed data using scaling techniques, (4) selection and evaluation of suitable remotely sensed data based on the scene model, (5) selection of suitable spatial analytic techniques to meet information requirements, and (6) cost-benefit analysis. Results from a case study show that the framework provided an objective mechanism to identify relevant aspects of the monitoring problem and environmental characteristics for selecting remotely sensed data and analysis techniques.
Stuart Kauffman here presents a brilliant new paradigm for evolutionary biology, one that extends the basic concepts of Darwinian evolution to accommodate recent findings and perspectives from the fields of biology, physics, chemistry and mathematics. The book drives to the heart of the exciting debate on the origins of life and maintenance of order in complex biological systems. It focuses on the concept of self-organization: the spontaneous emergence of order widely observed throughout nature. Kauffman here argues that self-organization plays an important role in the emergence of life itself and may play as fundamental a role in shaping life's subsequent evolution as does the Darwinian process of natural selection. Yet until now no systematic effort has been made to incorporate the concept of self-organization into evolutionary theory. The construction requirements which permit complex systems to adapt remain poorly understood, as is the extent to which selection itself can yield systems able to adapt more successfully. This book explores these themes. It shows how complex systems, contrary to expectations, can spontaneously exhibit stunning degrees of order, and how this order, in turn, is essential for understanding the emergence and development of life on Earth. Topics include the new biotechnology of applied molecular evolution, with its important implications for developing new drugs and vaccines; the balance between order and chaos observed in many naturally occurring systems; new insights concerning the predictive power of statistical mechanics in biology; and other major issues. Indeed, the approaches investigated here may prove to be the new center around which biological science itself will evolve. The work is written for all those interested in the cutting edge of research in the life sciences.
A holon is any stable sub-whole in a hierarchy. It is a self-creating, open system, governed by a set of laws which regulate its coherence, stability, structure and functioning. It also possesses the potential of adaptation to the challenge of environmental change. It examines interactions between levels in the landscape 'holarchy'. These are evaluated in terms of scale, communication, stability and evolution in a fluctuating energy stream. -from Author
In this paper a measure of the entropy associated with a channel network is defined, according to the Shannon informational definition, as the expectation of (-log Pj), where pj is the ratio of the existing paths at the bifurcation level j to the total number of paths. Here j and pj are respectively proportional to the arrival time of water to the network outlet and to the number of water parcels arriving from the distance j. Then, the expression for the channel network entropy proposed in this paper is well suited for hydrologic purposes. By analyzing a river network (and related sub-networks) of an Italian basin with surface area of 123 km2, it is shown that the network entropy is strictly related to basin characteristics such as average elevation, Horton order, and magnitude.
Nonlinear dynamical systems models of soil systems suggest that soil evolution may be potentially chaotic, and thus sensitive to initial conditions and to small perturbations. This hypothesis was tested by comparing the diversity of soil types on either side of the Suffolk Scarp on the lower Coastal Plain of North Carolina, the Pamlico Terrace and the Talbot Terrace. A soil system model based on vertical clay distribution and textural differentiation is introduced to account for the major factors that lead to differentiation of soil series in the study area. Only one soil series was indentified on the Pamlico site, while the Talbot transect included at least seven distinct soil series. The dramatically higher variability of the soil cover on the older landscape is predicted by the model and provides field evidence for chaotic pedogenesis. -from Author
The customary view of the landscape and of landscape processes emphasizes regularity. Macroscopic landforms are represented as contoured surfaces and mean values are held to provide a satisfactory knowledge of structures and processes at the micro level. However, it is known that geomorphic structures can exhibit a spatial variability that renders them unmappable and non-laminar flows are so complex as to defy regular analysis. Irregularity in the landscape may be probabalistic in origin or the outcome of non-linearity where, despite being completely determined, systems can exhibit stochasticity. Two examples of probabilistic irregularity are presented in this paper: (i) the soil-covered landscape as a sample function of a Gaussian field; (ii) variable thresholds as the subtraction of random variables. Non-linear stochasticity enters geomorphology mainly via dissipative structures in turbulent flow. For slightly perturbed, mildy non-linear, conservative systems, stability is provided for by KAM theory, while 'Hamiltonian chaos' guarantees ergodicity and the possibility of equilibrium.
Real world ecosystems (as opposed to their mathematical counterparts) are often enormously complex associations of species which interact in diverse ways. As a matter of practical necessity, field ecologists can rarely specify, much less quantify, all of the interactions. Consequently, empirically derived equations purporting to describe the dynamics of such systems generally consider fewer than the total number of interacting species. The present paper calls attention to this reduction in dimensionality and explores some of its consequences. In particular, attention is called to what are termed the Abstracted Growth Equations, those of reduced dimensionality, and to the way that these expressions derive from the underlying n-variable equations. The degree to which the Abstracted Equations accurately describe the dynamics of the species of interest is shown to depend on the time scale of these species relative to that of the species which are omitted. A general result relating the product of the eigenvalues of the Abstracted Equations to the corresponding product for the n-variable equations is proved. It is further pointed out that the distinction between Abstracted and n-variable equations suggests experiments which at least in principle should enable the empiricist to estimate the importance of species and interactions which are omitted. The relationship between Abstracted and n-variable equations is also discussed with regard to measuring competition coefficients and related parameters, and also to the problem of determining whether or not higher order interactions are present in laboratory microcosms. The analysis concludes by comparing the stability properties of several simplified models of community interactions with those of the corresponding one-species Abstracted Equations. It is shown, for the case of difference equations, in particular, that analysis of the one-species models may often lead one to conclude that the system is stable, whereas in fact it is unstable due to overdamping. The final Discussion relates the results of the present paper to previous studies that anticipate the view presented here, and comments on the quarrel that has developed between those ecologists who believe in the existence of community-wide patterns of body size and the like and those who reject this view. It is suggested that the resolution of this dispute may depend on our ability to classify subsystems of species (i.e., guilds) with regard to the extent to which their internal organization is influenced by variation in the larger communities in which they are embedded. Finally, it is shown that Roughgarden's principal results for community coevolution can be deduced from the Abstracted Growth Equations of a particular subset of the entire community.
The spatial structure of soil variability at the landscape scale was examined on adjacent geomorphic surfaces dating from 80 to 200 ka in eastern North Carolina. The purpose was to determine whether there is evidence at broader scales (distances of 102–104 m) for the divergent evolution observed in the field at very detailed scales (distances of 100–102 m). The state probability function (SPF), which measures spatial dependence for categorical environmental data along a transect, was applied to soil series mapped at a 1:24,000 scale. The older Talbot Terrace and younger Pamlico Terrace surfaces showed distinctly different patterns of spatial variability. The range of spatial dependence was shorter on the older surface (about 200 vs. 300 m), and the SPF was higher at any given distance, indicating more variability. The SPF for the Pamlico surface also indicates a periodicity related to fluvial dissection of the landscape, which is not readily detectable on the Talbot transect despite its greater degree of dissection. The results confirm earlier field studies which suggest that pedogenesis is marked by divergence, whereby differences in initial conditions or local perturbations persist and increase to produce a more variable soil cover.
We postulate that a basic process in nature operates on a minimal time-scale, Δ0>0, and that on this time-scale the dynamics is governed by a deterministic, discrete-time nonlinear transformation τΔ0:X→X, which is manifested as a (chaotic) process τΔ=τΔ0N on a larger (observable) time-scale Δ=NΔ0. We assume that τΔ possesses an observable measure μΔ and define the average energy of the dynamical system (X,τΔ,μΔ) bym2∫XτΔ(x)−xΔ2dμ,where m denotes mass if the process represents the motion of a particle. With this definition, we prove a conservation of energy principle for each Δ. In the case where Δ can be made arbitrarily small, this definition reduces to the classical definition of average energy for a mechanical system governed by a differential equation. Let EΔ and IΔ denote the energy and information content of the dynamical system observed at the time-scale Δ. Using elementary concepts from dynamical systems, it is shown that energy and information are related by EΔ=KIΔ2, where K is a constant. This allows the estimation of the information content of matter.
Cellular automaton models have enjoyed popularity in recent years as easily constructed models of many complex spatial processes, particularly in the natural sciences, and more recently in geography also. Most such models adopt a regular lattice (often a grid) as the basis for the spatial relations of adjacency that govern evolution of the model. A number of variations on the cellular automaton formalism have been introduced in geography but the impact of such variations on the likely behavior of the models has not been explored. This paper proposes a method for beginning to explore these issues and suggests that this is a new approach to the investigation of the relationships between spatial structure and dynamics of spatial processes. A framework for this exploration is suggested, and details of the required methods and measures are provided. In particular, a measure of spatial pattern—spatial information—based on entropy concepts is introduced. Initial results from investigation along the proposed lines are reported, which suggest that a distinction can he made between spatially robust and fragile processes. Some implications of this result and the methodology presented are briefly discussed.
Scale issues are often addressed in contemporary geo-ecological studies and form one of the major challenges in the fields of physical geography, hydrology and ecology. In this paper the application of hierarchy theory and response units is proposed as an approach towards scale-transcending environmental studies on degradation and geomorphological development.
Goals of the research were to establish which processes are important at what spatio-temporal scale, how hydro-geomorphological response is influenced by biological processes and whether hierarchy theory and the response unit approach can be used as an up-scaling methodology.
Results from two climatologically and geomorphological different regions are discussed, one dominated by water shortage (SE Spain) and the other by water surplus (Luxembourg). In both cases detailed process research was carried out at scales ranging from the micro-plot to the catchment. Process research was concentrated on understanding and quantifying sediment and water transfer through the geo-ecosystems studied.
Outcomes showed that in both cases the role of biological processes was important in the hydrological and degradation response of both areas. This was not only true for the finest scale levels but also had its impact on the emerging properties and response at the hillslope and catchment level.
Connectivity of runoff-generating and runoff-absorbing areas was important on all scale levels. Connectivity is dominated by both the rainfall magnitude–frequency–duration characteristics and physically and biologically controlled thresholds, which range from initial soil moisture contents, vegetation patterns or soil biological activity, to the presence of water harvesting structures.
The complex interrelationships of the processes involved showed that linear up-scaling from fine to broad scale is impossible, as many thresholds and non-linear processes are involved at specific scales. The identified response units are used to integrate these complex relationships in a relatively manageable way, and may provide a useful framework for up-scaling, and for understanding catchment hydro-geomorphological response and development. Copyright
Soil variation has often been considered to be composed of‘functional’ or ‘systematic’ variation that can be explained, and random variation (‘noise’) that is unresolved. The distinction between systematic variation and noise is entirely scale dependent because increasing the scale of observation almost always reveals structure in the noise. The white noise concept of a normally distributed random function must be replaced to take into account the nested, autocorrelated and scale‐dependent nature of unresolved variations. Fractals are a means of studying these phenomena. The Hausdorff‐Besicovitch dimension D is introduced as a measure of the relative balance between long‐ and short‐range sources of variation; D can be estimated from the slope of a double logarithmic plot of the semivariogram. The family of Brownian linear fractals is introduced as the model of ideal stochastic fractals.
Data from published and unpublished soil studies are examined and compared with other environmental data and simulated fractional Brownian series. The soil data are fractals because increasing the scale of observation continues to reveal more and more detail. But soil does not vary exactly as a Brownian fractal because its variation is controlled by many independent processes that can cause abrupt transitions or local second order stationarity. Estimates of D values show that soil data usually have a much higher proportion of short‐range variation than landform or ground water surfaces. The practical implication is that interpolation of soil property values based on observations from single 10 cm auger observations will be unsatisfactory and that some method of bulking or block kriging should be used whenever longrange variations need to be mapped.
Some possible approaches to the aggregation and disaggregation of soil data and information are presented as an opener to the more detailed discussion. The concepts of hierarchy, grain, extent, scale and variability are discussed. Slight modifications to the Hoosbeek-Bryant scheme to deal with spatial and temporal scales and various types of quantitative models are suggested. Approaches to aggregation or upscaling are reviewed. The contributions of representative elementary volume (REV), variograms, fractal theory, multi-resolution analysis using wavelets, critical point phenomena, renormalisation groups and transfer functions are discussed followed by a brief presentation of some ecological approaches including extrapolation by lumping, extrapolation by increasing model extent and extrapolation by explicit integration. A clear distinction must be made between additive and non-additive variables. The scaling of the former is much less problematic than the latter. Corroboration of any approach by testing against the aggregated values seems problematic. Methods of disaggregation or downscaling including transfer functions, mass-preserving or pycnophylactic methods are also discussed. In order to make quantitative advances, nested sampling or reanalysis of data in land information systems to obtain variance information over a complete range of scales is required. Finally an appeal is made for work to begin on a quantitative scale-explicit theory of soil variation.
Individual trees may have significant impacts on soil morphology. If these impacts are non-random such that some microsites are repeatedly preferentially affected by trees, complex local spatial variability of soils would result. A model of self-reinforcing pedologic influences of trees (SRPIT) is proposed to explain patterns of soil variability in the Ouachita Mountains, Arkansas. SRPIT postulates that trees are preferentially established on patches that are nutrient-rich and rock fragment poor relative to adjacent sites. The biomechanical effects of trees on soil, and decomposition of roots then maintain and reinforce the rock fragment and nutrient differences relative to surrounding soils, increasing the likelihood of successful future tree establishment. The links hypothesized in the SRPIT model are dynamically unstable, which would be necessary for the self-reinforcing mechanisms to operate. Soil variability in 16 study plots is dominated by local, within-plot variability, pointing to highly localized biological effects and consistent with the SRPIT model. Within each 0.127 ha plot, 4–11 different series, and 4–9 different rock fragment classes were found. Of the 10 paired pits at each plot, 3–7 pairs had different series in pits typically less than 1 m apart. On average, each of the 16 plots had 6.3 different soil types, 6 different rock fragment classes, and 60% of the sample pairs differing in soil series. Richness–area analysis of soil series, and of rock fragment classes, both indicate that pedodiversity is dominated by within-plot rather than between-plot variability. The vertical variations in the concentration of rock fragments in 40 of 58 soil pits is consistent with redistribution of soil material by tree throw, and there is also evidence of rock fragment displacement by tree growth and deposition in stump holes. Overall, results suggest that soil morphological effects of individual trees are an important source of soil spatial variability in forests, and that such effects are non-random over time. Thus even relatively homogeneous areas may be characterized by tree-rich patches which support repeated generations of trees, and tree-poor patches which more rarely host trees.
An updated review (corresponding to the inaugural talk delivered at the The International Workshop on Classical and Quantum Complexity and Nonextensive Thermodynamics, Denton, TX, April 3–6, 2000) of nonextensive statistical mechanics and thermodynamics is colloquially presented. Quite naturally the possibility emerges for using the value of q−1 (entropic nonextensivity) as a simple and efficient manner to provide, at least for some classes of systems, some characterization of the degree of what is currently referred to as complexity (M. Gell-Mann, The Quark and the Jaguar, Freeman, New York, 1994). A few historical digressions are included as well.
Earth surface systems are controlled by a combination of global factors (laws, principles and relationships that apply everywhere and always) and local factors which are place- and time-contingent. Formal arguments, framed in the context of the state factor model of soils and ecosystems, show that where the scales of the global and local components are sufficiently different, global or local dynamics can be discovered without considering the other. However, the same arguments show that the total, aggregated system cannot be understood without accounting for both the local and global components. The system dynamics cannot be recovered from the global or local controls alone. Local forms of spatial analysis provide both the conceptual and operational means to frame global factors within the spatial and historical contingency expressed in local controls. Applications have been limited in environmental sciences, but the potential is high. Entropy-based approaches also hold promise and may allow estimates of the relative contributions of global and local controls. An example of application to forest succession in the southeastern US coastal plain is designed to examine the relative importance of laws governing successional pathways with or without regular fires and of the local spatial and temporal contingencies, which control fire regime. The analysis shows that global laws governing forest succession account for about 80% and local contingencies associated with fire frequency for about 20% of the system connectance entropy.
Coastal dune systems are studied at time scales from seconds to millennia, and space scales from millimeters to kilometers. Present approaches to the study of coastal dunes make it difficult to integrate models and interpretations of these systems over these scale ranges and arrive at reasonable conclusions. It is argued that identification of key controls on dune development, measurement of those controls, and synthesis of data, describing past and present conditions and used as calibration points, will improve the viability of coastal dune models. Theoretical and empirical advances are necessary to improve the reliability of predictions across a range of geomorphological scales.Attempts at linking theoretical (systematic) models, process (synoptic) measurement, and historical or paleoenvironmental (synthetic) approaches make explicit the recognition that at time scales of more than a few hours, and space scales greater than a few hundred meters, deterministic models become unstable vis-à-vis prototype environments. Process “climatologies” provide one means to link process-based work with broader-scaled analysis. As scales increase, such climatologies will become less appropriate as the data become less reliable, or as the systems change, or as the scales become too large relative to the process record lengths. In these conditions, specific data (i.e. samples) representing points in time and space become check points for calibrating models. It should be possible, ideally, to integrate both up and down time and space scales. This is not yet possible.
Distinctions between cause and effect in landform development depend on the span of time involved and the size of the geomorphic system under consideration. Depending upon the temporal and areal frame of reference, variables such as channel morphology may be either dependent or independent. In terms of geologic time, landforms represent a stage in an erosion cycle and are dependent on time. On a short-term basis, components of geomorphic systems may be regarded as systems in dynamic equilibrium or in a steady state and are independent of time.
Some earth surface systems apparently exhibit deterministic chaos, where small differences in initial conditions produce increasingly divergent results. This casts doubt on the venerable concept of equifinality, whereby surface features converge to similar forms. However, chaotic systems exhibit a broader-scale order, and their complex patterns occur within well-defined limits. This broad-scale order arising from smaller-scale chaos produces simplexity, where simple rules and regularities emerge from underlying complexity, when the broad-scale structures are independent of the fine-scale details. Chaos precludes equifinality, sensu stricto, at certain scales, but simplexity produces equifinality at others. Simplexity is illustrated in a case study of soil-landform relationships.
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It is argued that the aggregation approach towards macroscale hydrological modelling, in which it is assumed that a model applicable at small scales can be applied at larger scales using ‘effective’ parameter values, is an inadequate approach to the scale problem. It is also unlikely that any general scaling theory can be developed due to the dependence of hydrological systems on historical and geological perturbations. Thus a disaggregation approach to developing scale-dependent models is advocated in which a representation of the distribution of hydrological responses is used to reflect hydrological heterogeneity. An appropriate form of distribution may vary with both scale and environment. Such an approach is dependent on the data available to define and calibrate the chosen subgrid parameterization. A parameterization based on a minimum patch representation is suggested and the problems of identification at the larger scale discussed.
1. A view of river basins 2. Fractal characteristics of river basins 3. Multifractal characteristics of river basins 4. Optimal channel networks: minimum energy and fractal structures 5. Self-organized fractal river networks 6. On landscape self-organization 7. Geomorphological hydrologic response 8. References.
The sensitivity of soil landscapes to climatic variability and hydroclimatic events can be expressed as a landscape change safety factor, the ratio of potential disturbance to resistance to change. The use of a geographic information system (GIS) enables the spatially-explicit modeling of landscape sensitivity, but also raises the risk of violating the characteristic scales of disturbance and resistance, because the GIS technically simplifies the extrapolation of models, and associated concepts, to landscapes and scales not represented by the digital data base. Embedding landscape sensitivity into hierarchy theory, the formal analysis of the hierarchical structure of complex systems, provides a conceptual framework for the transfer of models and variables among landscape scales. In the subhumid southern Canadian plains, major hydroclimatic events (strong winds, intense rain, rapid snow melt) cause much of the physical disturbance of soil landscapes and terrestrial ecosystems. Prolonged dry or wet weather influences the resistance of soil and vegetation to these events. The potential disturbance of soil landscapes therefore can be derived from the probabilities of extreme events and seasonal conditions, as recorded in instrumental and proxy climate records. This time series analysis can be linked to the modeling of landscape sensitivity by establishing the probabilities of hydroclimatic events and climatic conditions which may exceed or lower the resistance of individual soil landscapes.
The hierarchy theory framework is now widely used; some recent examples include Cammeraat A conceptually and operationally similar framework has devel-oped independently in pedology, based on Hoosbeek and Bryant (1992); see, for example
- Phinn
The hierarchy theory framework is now widely used; some recent examples include Cammeraat (2002) and Phinn et al. (2003). A conceptually and operationally similar framework has devel-oped independently in pedology, based on Hoosbeek and Bryant (1992); see, for example, McBratney (1998).
- M L Zdenkovic
- A E Scheidegger
Zdenkovic, M. L., and A. E. Scheidegger. 1989. Entropy of landscapes. Zeitschrift fur Geomorphologie
33:361–71.
Annals of the Association of American Geographers
- G P Malanson