Soil production function refers to the rate of bedrockweathering into soil as a function of soil thickness. A general model suggests that the rate of physical weathering of bedrock (de/dt) can be represented as an exponential decline with soil thickness:
where h is soil thickness [m], P0 [mm/year] is the potential (or maximum) weathering rate of bedrock and k [m−1] is an empirical constant.[1]
The reduction of weathering rate with thickening of soil is related to the exponential decrease of temperature amplitude with increasing depth below the soil surface, and also the exponential decrease in average water penetration (for freely-drained soils). Parameters P0 and k are related to the climate and type of parent material. The value of P0 was found to range from 0.08 to 2.0 mm/yr for sites in northern California, and 0.05–0.14 mm/yr for sites in southeastern Australia.[2] Meanwhile values of k do not vary significantly, ranging from 2 to 4 m−1.
Several landscape evolution models have adopted the so-called humped model.[3] This model dates back to G.K. Gilbert'sReport on the Geology of the Henry Mountains (1877). Gilbert reasoned that the weathering of bedrock was fastest under an intermediate thickness of soil and slower under exposed bedrock or under thick mantled soil. This is because chemical weathering requires the presence of water. Under thin soil or exposed bedrock water tends to run off, reducing the chance of the decomposition of bedrock.
Ahnert, F. (1977). "Some comments on the quantitative formulation of geomorphological process in a theoretical model". Earth Surface Processes. 2 (2–3): 191–201. doi:10.1002/esp.3290020211.
The exponential function is a mathematical function denoted by or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments , including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
A logistic function or logistic curve is a common S-shaped curve with the equation
Soil formation, also known as pedogenesis, is the process of soil genesis as regulated by the effects of place, environment, and history. Biogeochemical processes act to both create and destroy order (anisotropy) within soils. These alterations lead to the development of layers, termed soil horizons, distinguished by differences in color, structure, texture, and chemistry. These features occur in patterns of soil type distribution, forming in response to differences in soil forming factors.
Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent. Exponential growth is the inverse of logarithmic growth.
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival?
Chernozem, also called black soil, regur soil or black cotton soil, is a black-colored soil containing a high percentage of humus and high percentages of phosphorus and ammonia compounds. Chernozem is very fertile soil and can produce high agricultural yields with its high moisture storage capacity. Chernozems are a Reference Soil Group of the World Reference Base for Soil Resources (WRB)
A soil horizon is a layer parallel to the soil surface whose physical, chemical and biological characteristics differ from the layers above and beneath. Horizons are defined in many cases by obvious physical features, mainly colour and texture. These may be described both in absolute terms and in terms relative to the surrounding material, i.e. 'coarser' or 'sandier' than the horizons above and below.
In physical systems, damping is the loss of energy of an oscillating system by dissipation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include viscous damping in a fluid, surface friction, radiation, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes. Damping is not to be confused with friction, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping.
In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp(−t/τ).
The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). This function derives the formula for yx+b^x with logarithm. Dr. Allen is a key import in this equation. It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. The right-side or future value asymptote of the function is approached much more gradually by the curve than the left-side or lower valued asymptote. This is in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regard to detailing populations.
A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert Malthus, who wrote An Essay on the Principle of Population (1798), one of the earliest and most influential books on population.
The stretched exponential function
The early concepts of soil were based on ideas developed by a German chemist, Justus von Liebig (1803–1873), and modified and refined by agricultural scientists who worked on samples of soil in laboratories, greenhouses, and on small field plots. The soils were rarely examined below the depth of normal tillage. These chemists held the "balance-sheet" theory of plant nutrition. Soil was considered a more or less static storage bin for plant nutrients—the soils could be used and replaced. This concept still has value when applied within the framework of modern soil science, although a useful understanding of soils goes beyond the removal of nutrients from soil by harvested crops and their return in manure, lime, and fertilizer.
Terra rossa is a well-drained, reddish, clayey to silty soil with neutral pH conditions and is typical of the Mediterranean region. The reddish color of terra rossa is the result of the preferential formation of hematite over goethite. This soil type typically occurs as a discontinuous layer that ranges from a few centimeters to several meters in thickness that covers limestone and dolomite bedrock in karst regions. The high internal drainage and neutral pH conditions of terra rossa are a result of the karstic nature of the underlying limestone and dolomite. Terra rossa is also found associated with Mediterranean climates and karst elsewhere in the world.
Biological neuron models, also known as spiking neuron models, are mathematical descriptions of neurons. In particular, these models describe how the voltage potential across the cell membrane changes over time. In an experimental setting, stimulating neurons with an electrical current generates an action potential, that propagates down the neuron's axon. This axon can branch out and connect to a large number of downstream neurons at sites called synapses. At these synapses, the spike can cause release of a biochemical substance (neurotransmitter), which in turn can change the voltage potential of downstream neurons, potentially leading to spikes in those downstream neurons, thus propagating the signal. As many as 85% of neurons in the neocortex, the outermost layer of the mammalian brain, consists of excitatory pyramidal neurons, and each pyramidal neuron receives tens of thousands of inputs from other neurons. Thus, spiking neurons are a major information processing unit of the nervous system.
In statistics, the Matérn covariance, also called the Matérn kernel, is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis on metric spaces. It is named after the Swedish forestry statistician Bertil Matérn. It specifies the covariance between two measurements as a function of the distance between the points at which they are taken. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the Matérn covariance is also isotropic.
Plant litter is dead plant material that have fallen to the ground. This detritus or dead organic material and its constituent nutrients are added to the top layer of soil, commonly known as the litter layer or O horizon. Litter is an important factor in ecosystem dynamics, as it is indicative of ecological productivity and may be useful in predicting regional nutrient cycling and soil fertility.
The soil biomantle can be described and defined in several ways. Most simply, the soil biomantle is the organic-rich bioturbated upper part of the soil, including the topsoil where most biota live, reproduce, die, and become assimilated. The biomantle is thus the upper zone of soil that is predominantly a product of organic activity and the area where bioturbation is a dominant process.
Erodability is the inherent yielding or nonresistance of soils and rocks to erosion. A high erodability implies that the same amount of work exerted by the erosion processes leads to a larger removal of material. Because the mechanics behind erosion depend upon the competence and coherence of the material, erodability is treated in different ways depending on the type of surface that eroded.
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