Statistical properties of fast Fourier transform coefficients computed from real-valued, covariance-stationary, periodic random sequences

Fast fourier transforms 2-3-5

The application of spectral simulation is gaining acceptance because it honors the spatial distribution of petrophysical properties, such as reservoir porosity and shale volume. While it has been widely assumed that spectral simulation will reproduce the mean and variance of the important properties such as the observed net/gross ratio or global average of porosity, this paper shows the traditional way of implementing spectral simulation yields a mean and variance that deviates from the observed mean and variance. Some corrections (shift and rescale) could be applied to generate geologic models yielding the observed mean and variance; however, this correction implicitly rescales the input variogram model, so the variogram resulting from the generated cases has a higher sill than the input variogram model. Therefore, the spectral simulation algorithm cannot build geologic models honoring the desired mean, variance, and variogram model simultaneously, which is contrary to the widely accepted assumption that spectral simulation can reproduce all the target statistics. However, by using Fourier transform just once to generate values at all the cells instead of visiting each cell sequentially, spectral simulation does reproduce the observed variogram better than sequential Gaussian simulation. That is, the variograms calculated from the generated geologic models show smaller fluctuations around the target variogram. The larger the generated model size relative to the variogram range, the smaller the observed fluctuations.

Reproduction of the Mean, Variance, and Variogram Model in Spectral Simulation

Fracture set properties such as orientation, spacing, trace length, and waviness tend to be spatially correlated. These properties can be efficiently simulated by spectral analysis procedures that take advantage of the computational speed of the fast Fourier transform. The covariance function of each property to be simulated is obtained from the variogram function estimated from mapped fracture set data and is typically referenced to the mean vector of the set. Simulation procedures for normally and exponentially distributed data involve generating uncorrelated Fourier coefficients that are assigned proper variance according to the spectral density, which is the Fourier transform of the covariance function. These coefficients are then reverse Fourier transformed to produce simulated set properties that have the desired variance and variogram function.

Spectral-Type Simulation of Spatially Correlated Fracture Set Properties ~

Geostatistical simulation is used to create a synthetic three‐dimensional geologic environment such that randomly generated values of a physical property have a required covariance and mean. We create a spatially stationary, multivariate Gaussian simulation with a Gaussian covariance. This type of simulation permits modeling of natural variations which may be an important source of noise in geologic investigations, whereas interpolation techniques such as kriging tend to smooth out those variations. Computationally efficient frequency‐domain methods allow the rapid creation of multiple scenarios, each having the required statistical distribution. For the particular problem considered, a given porosity distribution is simulated and then functionally related to physical properties needed for input into geophysical models. Additionally, megascopic cavities having random size, shape, and orientation similar to those observed in karst settings are emplaced in the physical model. Geostatistical simulation plays...

Statistical properties of fast Fourier transform coefficients computed from real-valued, covariance-stationary, periodic random sequences

Citations

Fast fourier transforms 2-3-5

Reproduction of the Mean, Variance, and Variogram Model in Spectral Simulation

Spectral-Type Simulation of Spatially Correlated Fracture Set Properties ~

Geostatistical simulation for geophysical applications Part I: Simulation

Conditional Simulations of Ocean Wave Properties

Related Papers (5)

On the representation of sequences as Fourier coefficients

κ-deformed Fourier transform

On Symmetric Alpha-Stable Noise After Short-Time Fourier Transformation

On the residual correlation of finite-dimensional discrete Fourier transforms of stationary signals (Corresp.)

The discrete Fourier transform approximation for periodically correlated time series

Statistical properties of fast Fourier transform coefficients computed from real-valued, covariance-stationary, periodic random sequences

Citations

Fast fourier transforms 2-3-5

Reproduction of the Mean, Variance, and Variogram Model in Spectral Simulation

Spectral-Type Simulation of Spatially Correlated Fracture Set Properties ~

Geostatistical simulation for geophysical applications­ Part I: Simulation

Conditional Simulations of Ocean Wave Properties

Related Papers (5)

On the representation of sequences as Fourier coefficients

κ-deformed Fourier transform

On Symmetric Alpha-Stable Noise After Short-Time Fourier Transformation

On the residual correlation of finite-dimensional discrete Fourier transforms of stationary signals (Corresp.)

The discrete Fourier transform approximation for periodically correlated time series

Geostatistical simulation for geophysical applications Part I: Simulation