Models of Landscape Evolution

Summary: Intellectual structures are critical in geomorphology because

1. landforms develop over longer timespans (usually much longer) than human lifespans
   
   
   • therefore we resort to analogues either in the present or historical records to explain the evolution of landscapes
   • the extrapolation of short records over long timespans requires a theoretical basis
   
   
2. there are different methodologies and objectives depending on the relevant theory
   
   
   • this is most apparent in the contrast between research which links characteristic form to present day processes (and thus limits retrodiction and prediction) versus research that focuses on relaxation forms but does not address relationships between contemporary forms and processes
   • appreciation of theory leads to recognition that there is an artificial entry point to this circular argument and that the starting point will often depend on the nature of a landscape and objectives of the study
   
   
3. by its vary nature, erosion eliminates evidence of landscape evolution
   
   
   • thus we depend very much on depositional records (stratigraphy)
   • however rarely are depositional records continuous and gaps present significant challenges and opportunities for interpretation
   • just as we need a strong theoretical basis for extrapolating from short process records, the same applies to the use of fragmented depositional records
   
   
4. the most profitable approach to landform studies is to derive hypotheses from process research and corroborate or falsify them with morphometric research
   
   
   • however for logistical and methodological reasons, process studies are local
   • once again a conceptual framework is required to link scales and permit the testing of hypotheses about landscapes
   
   

W.M. Davis: The geographical cycle

• the first model of landscape evolution to gain widespread acceptance within the discipline
• was remarkably influential and persistent but no longer dominates research thinking like it did, but still used as a teaching tool and residual influence reflected in the way geomorphologists cling to cyclical models

Thus Davis aspired to a deductive, theoretical, genetic model of landscape evolution. The concepts of structure, process and time were his theoretical framework:

• structure was regional and considered as an initial condition (beyond the scope of his model)
• process was the sum of weathering and transport rather than specific processes or mechanisms, although since his cycle was based on the assumption of a normal climate, i.e. humid temperate, fluvial processes predominate
• time was the central theme, but time in the sense of landscape development relative to the completion of the entire geographical cycle, i.e.,extent of landscape development or stage

Walter Penck: relating landforms to crustal movements

• similar to Davis in terms of morphological and deductive but failure to relate hypothesized forms and field observations
• but whereas Davis related form to stage, Penck related it to rate of uplift and he rejected Davis' assumption that uplift is followed by erosion of a stable crust
• he didn't see a sequence of landform development but rather various possible sequences according to differing rates of uplift and erosion
• he was careful to define the domain of his model as sub-aerial, but excluding eolian and glacial processes and climatic variability

Another expression of his model was three categories of landform assemblage according to tectonic history (versus normal climate):

• great folding from lateral forces (orogenic)
• dome formation without folding (epeirogenic)
• stable regions

Evaluation of Penck's model is hindered by its hurried writing, posthumous publication and confused representation in English, including misrepresentation by Davis who was defending his own ideas. Although there were important flaws and contradictions in Penck's work, and it was poorly translated or misrepresented especially by adherents to the Davisian school, it was the one comprehensive alternative to the cycle of erosion and thus was a focus for contrary ideas, such as emphasis on process rate (both endogenic and exogenic) and greater attention to slope retreat.

Lester King: global geomorphology during the era of process studies

• King was trained in the Davisian school but eventually rejected many of Davis' notion as he attempted to apply the cycle of erosion to an interpretation of the landscape of South Africa
• his model is based on the notion of parallel retreat of scarps and the resulting constraint on the downwearing of the surface of the scarp

1. waxing slope: convex segment at the crest, dominated by soil creep of a weathered mantle; increasing slope angle is required to transport the greater quantity of slope debris with distance downslope
2. free face: bedrock outcrop; retreats parallel with weathering and uniform removal; may be absent in areas of low relief
3. debris slope: debris from the free face resting at the angle of repose; does not bury the free face but retreats with it
4. waning: gentle concave profile controlled by sheetwash and transport of sediment over an eroded bedrock surface (pediment)

• the corresponding sequence of landscape evolution would be expansion and coalescence of pediments (pediplanation to form pediplains) as the free face retreats eliminating the higher surfaces (older pediments) and creating isolated erosional remnants
• King suggested that this evolution of landscapes would occur everywhere, but regarded the arid cycle as dominant given its extent geographically and over geological time
• his model implies that slopes lacking the free face (i.e., having the classic sigmoidal slope) are inactive and a degenerative from of his classic profile

Systems modelling

Systems modelling swept though geomorphology in the 1960's with the shift to process and quantitative geomorphology. Although not directly related to process geomorphology, the two methodologies are so compatible that they are closely associated in the same manner as Davis'' cycle of erosion and denudation chronology.

The formal structure of systems modelling (inputs, outputs, throughputs), and its application either conceptually or quantitatively make it a popular framework for research and teaching (hydrological cycle, continuity equations).

System

a set of objects together with relationships between the objects and their attributes, or a set of objects and the common processes by which they interact

The components of the system will depend on the relationships that the researcher or teacher considers important or relevant. A model is a simplification of a system that is assumed to exist in the real world. Since it will be based on selected parts of a system, the model will always be incorrect to some degree.

Scientists can agree on the definition of a particular system, but it can never be fully known. Conversely, a model can be simple and completely known, but always will be incomplete and useful only for specific purposes.

Types of systems:

1. isolated: assumed to have boundaries that prevent the import or export of energy and mass; besides the universe, the only isolated systems are in labs or assumed for convenience
2. closed: import or export of energy but not mass; the earth is closed to closed and some geomorphic systems can be modelled as if they are (e.g. an internal drainage basin)
3. open: both energy and mass move freely across system boundaries, includes most of the natural world

Types of system models according to complexity:

1. qualitative data, verbal description, graphs to describe a system; does not permit true modelling but is a first stage
2. identification and analysis of morphological variables (e.g. free face and talus); simplest model, only functional information is relative position of the components, but no inference or understanding of mass or energy exchange is possible
3. identification and analysis of energy and mass transfer; cascading system (direction specified; energy and mass move from the free face to the talus and not vice versa)
4. integration of form and process; unlike 3 these involve feedback, output from one part is input to another, opposing or slowing the general trend of the system (negative feedback) or magnifying the main trend (positive feedback); natural systems (except glacial systems) are characterized by negative feedback so that changes are resisted and tendency towards an equilibrium is maintained; thus change in relationships over time can be modelled, for example, the talus grows to bury the free face resulting in negative feedback and decreasing rates of mass wasting and talus slope growth over time
5. control systems: conscious or deliberate human intervention in the system; useful for addressing engineering or planning issues (impact assessment)

Types of system models according to degree of understanding of the system:

1. black box: nothing known except the relationship between input and output
2. grey box: structure is known (subsystems considered), but no detailed knowledge or investigation
3. white box: subsystems, storages and flows known and investigated in detail

Construction of a system model

• requires identification of the important morphological variables, and linkages among these in terms of direction and strength
• one of the simplest and most common quantitative techniques is correlation: that is establishing the covariation among pairs of variables using either nominal/ordinal data (non parametric) or interval/ratio data (parametric)
• correlation does not distinguish between dependent and independent variables because variance in both variables is examined; therefore cause and effect can only be determined by interpreting other evidence; linkages established in this manner represent only statistical relationships and no the flow of energy or matter
• the principal objective of science is to predict the behaviour of systems by determining the behaviour of individual components with absolute certainty using deterministic mathematics; this usually is not possible, so the behaviour of groups of objects is predicted with some known margin of error using the statistical methods of stochastic mathematics
• a fundamental between rational and empirical equations is that the former balance dimensionally, that is, when dimensions of mass, length, time and temperature are summed the units are either the same on both sides of an equation or result in dimensionless quantities, including angles; the advantage of deterministic modeling is prediction with certainty and dimensionless numbers apply to a range of scales, however energy and mass transfer models cannot be mathematically integrated given the dimensional incompatibility
• besides equations, relationships within and among subsystems are depicted with canonical structures usually as a schematic diagram as a geomorphic interpretation of a system model; these diagrams are able to show the nature (direct or inverse), direction (dependent/independent), and strength of linkages within and among subsets of variables; by following pathways it is possible to identify feedback
• the special significance of certain variables can be shown symbolically, in particular, regulators are variables that tend to stabilize the system internally (e.g. thresholds variables - slope angle, infiltration capacity; those that influence energy or mass transfer - vegetation; those that are important in terms of presence/absence - basal erosion), while stores are components that retain and then release energy, mass or information (e.g. vegetation, soil water, sediment sinks, energy sinks)

Evaluation of systems modelling

• the greatest strength is the consistent structure across a range of application, complexity and techniques
• the use of symbols is powerful means of simplification and abstraction and thereby enables the use of mathematical symbols
• greatest weakness if the choice was system components which is arbitrary and unscientific (there is no standard approach) and influences are further analysis and interpretation
• similar to the problem of distinguishing between core and peripheral variables is identifying the limits of the system, that is, defining boundaries
• systems models often are used just to present what is already known, that is, to summarize relationships by presenting them diagrammatically; this is problem of application and not the methodology; systems models should be related to theory by displaying new relationships and serving to generate hypotheses
• system models are abstractions not reality, they must be tied to theory and treated as simplifications with limitations

John T. Hack: A time-independent model

John T. Hack is the champion of a time-independent model where landscape variability due to age is not modelled, but rather considered a source of variability in landscape from related to contemporary process. This approach assumes a dynamic equilibrium between contemporary surficial processes and the surface upon which they are acting. Hack chose dynamic equilibrium as his conceptual and methodological framework. He derived this perspective directly from G.K. Gilbert who worked in the western US, where the dramatic semiarid landscape seems youthful and dynamic. Hack applied dynamic equilibrium to reinterpretation of the Appalachian Mountains, the landscape that lead Davis to think in terms of change over time.

The time-independent perspective of Gilbert is reflected in his "laws" of

• uniform slope: non-linear increase in rate of erosion with slope angle
• structure: differential erosion on resistant and non-resistant substrata
• divides of increasing acclivity: stream and slope gradients increase towards divides
• tendency of equality of action: same rates of erosion on hard and soft rocks through adjustment of slope angles; steep and high relief in strong rocks, low gradient and relief in weak rocks

The basic features of dynamic equilibrium as applied to spatial relations within a drainage basin by Hack:

• all elements of the topography are mutually adjusted so that they are downwasting at the same rate
• forms and processes are in a steady state of balance
• differences and characteristics of form are explainable in terms of spatial relations in which geologic patterns are the primary consideration rather than a theoretical evolutionary development
• opposing forces (inputs and outputs) are in a state of balance where their effects cancel out to produce a steady state

Evaluation of dynamic equilibrium:

1. the past usually is poorly or only partly known, thus a model based on current conditions has a definite advantage
2. mutual relationship with process geomorphology
3. time-independent is an end-member of the distribution of systems and system models; these are fairly easily identified (e.g. and underfit stream is time-dependent relative to valley form but time-independent with respect to channel form)
4. however, it is not usually this easy to resolve the complex forms that represent both time-independent and time-dependent behaviour; attention to spatial and temporal scale help, for example time-independent behaviour is more likely at more local scales, and the influence of past processes is proportional to their intensity and inversely proportional to time elapsed since the event
5. dynamic equilibrium implies characteristic forms as opposed to relaxation forms
6. situations where form is not maintained include uplift exceeding rates of erosion or increasing relief controlled by difference in rock resistance (e.g., inversion of topography)
7. When small segments of landscape evolution are sampled it becomes difficult or impossible to distinguish between dynamic equilibrium (trending mean), steady state (constant mean) and dynamic metastable equilibrium (two scales of oscillations). Hack referred to both steady state and dynamic equilibrium, however a trending mean is much more likely in geomorphic systems.
8. Dynamic equilibrium is more of a conceptual framework than a fully tested corroborated model, which will require much more extensive data bases. However, it is a very useful framework in that the reduced role of time is replaced with an expansion of spatial variability and the integration of parts of landscapes. In this respect it is tied to a systems approach and the notion that systems move toward equilibrium at a rate proportional to their distance from it. Thus those far from equilibrium change quickly (time-independent) and thus near equilibrium change slowly (time-dependent). This systems perspective unites both the time-dependent and time-independent viewpoints.

Mathematical Modelling

Mathematics allows ideas to be expressed with great precision but this also demands a great deal of knowledge about the phenomenon. With this precision and lack of ambiguity, mathematical models have internal logic (one step leads logically to another), and thus the primary source of error is external logic, i.e. the assumptions required so that the model can be expressed mathematically. In geomorphology, these assumption are often very constraining or even unrealistic given our incomplete understanding of geomorphic systems.

The use of mathematics provides the option of modelling which is deterministic (where one physical or chemical conditions leads uniquely from another) or probabilistic (a degree of randomness and therefore variability is introduced), with the recognition the natural world is deterministic but too complex to be modelled as such. Mathematical modelling has been out of the mainstream of geomorphological research with the emphasis on process observation and an inferential inductive methodology.

Following is a review of the mathematical models of two contemporary geomorphologists. The focus here is on methodology and logic rather than the mathematics.

Frank Ahnert: a mathematical model of slope evolution

• a German born and educated geomorphologist who was a professor at the University of Maryland for many years and so his approach to geomorphology was typical of the North American/ British school of process geomorphology

Ahnert's model is discrete: comprised of a network of points, where computations are made individually and sequentially with small temporal and spatial increments. Waste thickness is related to form by modelling changes along slope profiles (2-D) or on a grid of points (3-D). He differentiated between direct removal (no redeposition), and point-to-point transfer (the norm) which results in accumulation of debris at the foot of the slope. The computations are of the change in waste cover thickness (mass balance) in response to mechanical and chemical weathering, slope wash, rain splash, soil creep and landsliding. Rational equations simulate each process.

The model user is able to simulate the influence of climate by varying the relative importance of mechanical and chemical weathering. The dynamic equilibrium between waste thickness and rates of includes decreasing weathering as waste thickens and increasing weathering as the waster cover thins, although a thin waste cover is most effective for chemical weathering and so rate of weathering initially increases with increasing waste cover until a threshold thickness beyond which weathering rates decline.

Splash transport is a non-linear function of slope gradient, while wash transport is a non-linear function of slope gradient, surface runoff and soil erodibility. Wash transport can be specified as point-to point or complete removal. Surface runoff is derived from a slope hydrology submodel based on the mass balance between precipitation at a point, infiltration capacity and runoff from upslope.

Ahnert modelled slow mass movement as either viscous (nonthreshold) or plastic (threshold), depending on slope gradient and fluidity of the material. In both cases, there is a critical slope angle but the value is zero for nonthreshold viscous mass wasting. Beyond a threshold slope, waste does not accumulate, so Ahnert has the material move to the next lowest point where slope gradient is less than the threshold. However this not a realistic description of rapid mass wasting, because 1) momentum usually carries the debris further than this model suggests and 2) the critical threshold for rapid mass wasting will vary from point to point as kinetic energy is converted to kinetic energy (i.e., in a moving mass the angle of dynamic friction is less than the angle of static friction).

Ahnert approach to modelling is numerical simulation, that is, the outcome depends not just on the mathematical functions but also the numerical values that the user enters to simulate, for example, rock structure (resistance) at a point, nature of base level change (reference data for iterations of the model), relative importance of weathering processes and dominant process of mechanical transport. By varying all these boundary conditions, Ahnert concluded that the strongest control on slope profile form was the mode of removal (point-to-point or direct).

The greatest strengths of his model are the use of waste thickness as a control and response with respect to slope processes and as a parameter than can be verified empirically. Also his model is geographic; it can accommodate spatial variability between point locations. The two main weaknesses are that he does not model feedback among processes and some relationships are suggested by observation and not physically based.

Michael J. Kirkby: deterministic modelling of process-response

Kirkby slope model is based on a mass balance equation formulated using differential calculus so that, unlike Ahnert, the slope is modelled simultaneously as a continuous profile. It is structured to accept submodels and the analytical use is based on vectors, although a co-ordinate system must be specified in order to solve the equations. Kirkby models change in elevation as a function of mechanical and chemical transport processes.

Kirkby's initial and preferred approach was analytical, that is, expressing theory symbolically and deriving new equations by manipulating the symbols; very much a deductive methodology. He eventually resorted to numerical modelling for the modelling of complex problems where the theory is inadequate, however, once empirical data (numbers) are inserted into equations, only numbers can be derived and not new equations.

Kirkby based the simulation of weathering effects on the concept of a soil deficit:

• the amount of material required to reconvert soil into bedrock ("everywhere within a soil some material has been removed chemically in comparison to the bedrock from which it is derived")
• this soil profile model is primarily chemical in nature and strongly dependent on the effects of hillslope hydrology on chemical weathering and solute transport
• rate of change in chemical removal is constant with respect to changes in the volume of subsurface flow

• erosion-limited removal: from transport-limited where the actual transport rate is equal to the transporting capacity of the process versus weathering-limited where the capacity of the transporting process greatly exceeds the transport rate
• a hydrological submodel based on annual precipitation, mean rain per day, surface storage capacity and evapotranspiration combined in a mass balance
• treating landsliding as a threshold-based, erosion-limited process and including distance travelled by the landslide as a function of slope gradient