powerLawPDF: Solved probability functions

In FluvialLandscapeLab/hydrogeom: Calculate Hydrologic Geometry of Expansive Hyporheic Zones

Solved probability functions

Description

hydrogeom can provide transient storage zone statistics based on solutions to six integrations of a shape file. These functions are examples of solutions for the powerLaw and exponent shape functions.

Usage

powerLawPDF(tau, tau_0, tau_n, alpha)

powerLawCCDF(tau, tau_0, tau_n, alpha)

powerLawIntPDF(tau_a, tau_b, tau_0, tau_n, alpha)

powerLawIntCCDF(tau_a, tau_b, tau_0, tau_n, alpha)

powerLawIntTau.CCDF(tau_a, tau_b, tau_0, tau_n, alpha)

powerLawIntTau.PDF(tau_a, tau_b, tau_0, tau_n, alpha)

exponentPDF(tau, tau_0, tau_n, sigma)

exponentCCDF(tau, tau_0, tau_n, sigma)

exponentIntPDF(tau_a, tau_b, tau_0, tau_n, sigma)

exponentIntCCDF(tau_a, tau_b, tau_0, tau_n, sigma)

exponentIntTau.PDF(tau_a, tau_b, tau_0, tau_n, sigma)

exponentIntTau.CCDF(tau_a, tau_b, tau_0, tau_n, sigma)

Arguments

tau	A residence time for which the PDF or CCDF value is desired.
tau_0, tau_n	The total range of tau values considered.
alpha	The exponent of the power law (this value is negated within the functions; entering 1.5 yields a power law with an exponent of -1.5).
tau_a, tau_b	The lower and upper range for the definite integral of the PDF or CCDF.
sigma	The exchange rate of groundwater with an exponential distribution. (This value is negated in the calculations to simulate the decay of water volume with residence time.)

Details

Calculation of transient storage zone (TSZ) statistics (see TSZStats) requires six functions:

1) the probability density function (PDF) of the curve shape (i.e., the probability that a water molecule will exit the hyporheic zone with a residence time of tau)

2) the complementary cumulative distribution function (CCDF) (i.e., the probablity that a water molecule will still be in the hyporheic zone at residence time tau)

3) the definite integral of the PDF.

4) the definite integral of the CCDF.

5) the definite integral of tau*PDF.

6) the definite integral of tau*CCDF.

These values can be derived with numeric integration of a shape function (e.g., see powerLaw), but numerical stablity can be an issue (e.g., see checkShapeFunction). Providing solutions for the six integrations, above, may yield more reliable stability in the calculations.

The solutions are published in Poole et al. (in Press) and their implementation in this package can be viewed by printing the source code for each of the following functions (e.g., type print(<functionName>) in the R console).

Solution functions for other shapes can be provided by the user. However, they must follow the naming conventions of the concatination of the shape name and the specific suffixes: "PDF", "CCDF", "IntPDF", "IntCCDF", "IntTau.PDF", and "IntTau.CCDF". For instance, for a shape called "foo", the function names would be "fooPDF", "fooCCDF", "fooIntPDF", etc. Also, the PDF and CDF solutions must have the signature function(tau, tau_0, tau_n, ...) while the remaining solutions must have the signature function(tau, tau_0, tau_n, ...) where "..." are additional parameters required by the shape function.

Value

Returns a vector of numeric values.

PowerLawPDF and exponentPDF return the PDF value for tau. This provides the probability density that a water molecule entering the hyporheic zone at time == tau_0 will return to the stream at time == tau.

PowerLawCCDF and exponentCCDF return the CCDF value for tau. This provides the probablity that a water molecule entering the hyporehic zone at time == tau_0 will still be in the hyporheic zone at time tau (e.g., will upwell after time == tau)

PowerLawIntPDF and exponentIntPDF return the definite integral from tau_a to tau_b of the PDF.

PowerLawIntCCDF and exponentIntCCDF return the difinite integral from tau_a to tau_b of the CCDF.

PowerLawIntTau.PDF and exponentIntTau.PDF returns the definite integral from tau_a to tau_b of tau*PDF.

PowerLawIntTau.CCDF and exponentIntTau.CCDF return the difinite integral from tau_a to tau_b of tau*CCDF.

FluvialLandscapeLab/hydrogeom documentation built on May 13, 2022, 10 p.m.