gstools.covmodel.models¶

GStools subpackage providing different covariance models.

The following classes and functions are provided

`Gaussian`([dim, var, len_scale, nugget, …])	The Gaussian covariance model
`Exponential`([dim, var, len_scale, nugget, …])	The Exponential covariance model
`Matern`([dim, var, len_scale, nugget, anis, …])	The Matérn covariance model
`Rational`([dim, var, len_scale, nugget, …])	The rational quadratic covariance model
`Stable`([dim, var, len_scale, nugget, anis, …])	The stable covariance model
`Spherical`([dim, var, len_scale, nugget, …])	The Spherical covariance model
`Linear`([dim, var, len_scale, nugget, anis, …])	The bounded linear covariance model
`MaternRescal`([dim, var, len_scale, nugget, …])	The rescaled Matérn covariance model
`SphericalRescal`([dim, var, len_scale, …])	The rescaled Spherical covariance model

class gstools.covmodel.models.Gaussian(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The Gaussian covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \exp\left(- \frac{\pi}{4} \cdot \left(\frac{r}{\ell}\right)^2\right)$

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`calc_integral_scale`()	The integral scale of the gaussian model is the length scale
`correlation`(r)	Gaussian correlation function
`covariance`(r)	Covariance of the model
`spectral_rad_cdf`(r)	The cdf of the radial spectral density
`spectral_rad_ppf`(u)	The ppf of the radial spectral density
`spectrum`(k)	The spectrum of the covariance model.
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

calc_integral_scale()[source]¶: The integral scale of the gaussian model is the length scale

correlation(r)[source]¶: Gaussian correlation function

$\mathrm{cor}(r) = \exp\left(- \frac{\pi}{4} \cdot \left(\frac{r}{\ell}\right)^2\right)$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

spectral_rad_cdf(r)[source]¶: The cdf of the radial spectral density

spectral_rad_ppf(u)[source]¶: The ppf of the radial spectral density

spectrum(k)[source]¶

The spectrum of the covariance model.

This is given by:

$S(k) = \left(\frac{1}{2\pi}\right)^n \int C(r) e^{i b\mathbf{k}\cdot\mathbf{r}} d^n\mathbf{r}$

Internally, this is calculated by the hankel transformation:

$S(k) = \left(\frac{1}{2\pi}\right)^n \cdot \frac{(2\pi)^{n/2}}{(bk)^{n/2-1}} \int_0^\infty r^{n/2-1} f(r) J_{n/2-1}(bkr) r dr$

Where $C(r)$ is the covariance function of the model.

Parameters:	k (`float`) – Radius of the phase: $k=\left\Vert\mathbf{k}\right\Vert$

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

class gstools.covmodel.models.Exponential(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The Exponential covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \exp\left(- \frac{r}{\ell} \right)$

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`calc_integral_scale`()	The integral scale of the exponential model is the length scale
`correlation`(r)	Exponential correlation function
`covariance`(r)	Covariance of the model
`spectral_rad_cdf`(r)	The cdf of the radial spectral density
`spectral_rad_ppf`(u)	The ppf of the radial spectral density
`spectrum`(k)	The spectrum of the covariance model.
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

calc_integral_scale()[source]¶: The integral scale of the exponential model is the length scale

correlation(r)[source]¶: Exponential correlation function

$\mathrm{cor}(r) = \exp\left(- \frac{r}{\ell} \right)$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

spectral_rad_cdf(r)[source]¶: The cdf of the radial spectral density

spectral_rad_ppf(u)[source]¶: The ppf of the radial spectral density

spectrum(k)[source]¶

The spectrum of the covariance model.

This is given by:

$S(k) = \left(\frac{1}{2\pi}\right)^n \int C(r) e^{i b\mathbf{k}\cdot\mathbf{r}} d^n\mathbf{r}$

Internally, this is calculated by the hankel transformation:

$S(k) = \left(\frac{1}{2\pi}\right)^n \cdot \frac{(2\pi)^{n/2}}{(bk)^{n/2-1}} \int_0^\infty r^{n/2-1} f(r) J_{n/2-1}(bkr) r dr$

Where $C(r)$ is the covariance function of the model.

Parameters:	k (`float`) – Radius of the phase: $k=\left\Vert\mathbf{k}\right\Vert$

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

class gstools.covmodel.models.Spherical(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The Spherical covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \begin{cases} 1-\frac{3}{2}\cdot\frac{r}{\ell} + \frac{1}{2}\cdot\left(\frac{r}{\ell}\right)^{3} & r<\ell\\ 0 & r\geq\ell \end{cases}$

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`correlation`(r)	Spherical correlation function
`covariance`(r)	Covariance of the model
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

correlation(r)[source]¶: Spherical correlation function

$\mathrm{cor}(r) = \begin{cases} 1-\frac{3}{2}\cdot\frac{r}{\ell} + \frac{1}{2}\cdot\left(\frac{r}{\ell}\right)^{3} & r<\ell\\ 0 & r\geq\ell \end{cases}$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

class gstools.covmodel.models.SphericalRescal(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The rescaled Spherical covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \begin{cases} 1-\frac{9}{16}\cdot\frac{r}{\ell} + \frac{27}{1024}\cdot\left(\frac{r}{\ell}\right)^{3} & r<\frac{8}{3}\ell\\ 0 & r\geq\frac{8}{3}\ell \end{cases}$

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`calc_integral_scale`()	The integral scale of this spherical model is the length scale
`correlation`(r)	Rescaled Spherical correlation function
`covariance`(r)	Covariance of the model
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

calc_integral_scale()[source]¶: The integral scale of this spherical model is the length scale

correlation(r)[source]¶: Rescaled Spherical correlation function

$\mathrm{cor}(r) = \begin{cases} 1-\frac{9}{16}\cdot\frac{r}{\ell} + \frac{27}{1024}\cdot\left(\frac{r}{\ell}\right)^{3} & r<\frac{8}{3}\ell\\ 0 & r\geq\frac{8}{3}\ell \end{cases}$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

class gstools.covmodel.models.Rational(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The rational quadratic covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \left(1 + \frac{1}{2\alpha} \cdot \left(\frac{r}{\ell}\right)^2\right)^{-\alpha}$

$\alpha$ is a shape parameter and should be > 0.5.

Other Parameters:

**opt_arg – The following parameters are covered by these keyword arguments
alpha (float, optional) – Shape parameter. Standard range: (0, inf) Default: 1.0

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`correlation`(r)	Rational correlation function
`covariance`(r)	Covariance of the model
`default_opt_arg`()	The defaults for the optional arguments:
`default_opt_arg_bounds`()	The defaults boundaries for the optional arguments:
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

correlation(r)[source]¶: Rational correlation function

$\mathrm{cor}(r) = \left(1 + \frac{1}{2\alpha} \cdot \left(\frac{r}{\ell}\right)^2\right)^{-\alpha}$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

default_opt_arg()[source]¶

The defaults for the optional arguments:

{"alpha": 1.0}

Returns:	Defaults for optional arguments
Return type:	`dict`

default_opt_arg_bounds()[source]¶

The defaults boundaries for the optional arguments:

{"alpha": [0.5, inf]}

Returns:	Boundaries for optional arguments
Return type:	`dict`

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

class gstools.covmodel.models.Stable(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The stable covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \exp\left(- \left(\frac{r}{\ell}\right)^{\alpha}\right)$

$\alpha$ is a shape parameter with $\alpha\in(0,2]$

Other Parameters:

**opt_arg – The following parameters are covered by these keyword arguments
alpha (float, optional) – Shape parameter. Standard range: (0, 2] Default: 1.5

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`check_opt_arg`()	Checks for the optional arguments
`correlation`(r)	Stable correlation function
`covariance`(r)	Covariance of the model
`default_opt_arg`()	The defaults for the optional arguments:
`default_opt_arg_bounds`()	The defaults boundaries for the optional arguments:
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

check_opt_arg()[source]¶

Checks for the optional arguments

Warns:	alpha – If alpha is < 0.3, the model tends to a nugget model and gets numerically unstable.

correlation(r)[source]¶: Stable correlation function

$\mathrm{cor}(r) = \exp\left(- \left(\frac{r}{\ell}\right)^{\alpha}\right)$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

default_opt_arg()[source]¶

The defaults for the optional arguments:

{"alpha": 1.5}

Returns:	Defaults for optional arguments
Return type:	`dict`

default_opt_arg_bounds()[source]¶

The defaults boundaries for the optional arguments:

{"alpha": [0, 2, "oc"]}

Returns:	Boundaries for optional arguments
Return type:	`dict`

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

class gstools.covmodel.models.Matern(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The Matérn covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \frac{2^{1-\nu}}{\Gamma\left(\nu\right)} \cdot \left(\sqrt{2\nu}\cdot\frac{r}{\ell}\right)^{\nu} \cdot \mathrm{K}_{\nu}\left(\sqrt{2\nu}\cdot\frac{r}{\ell}\right)$

Where $\Gamma$ is the gamma function and $\mathrm{K}_{\nu}$ is the modified Bessel function of the second kind.

$\nu$ is a shape parameter and should be >= 0.5.

Other Parameters:

**opt_arg – The following parameters are covered by these keyword arguments
nu (float, optional) – Shape parameter. Standard range: [0.5, 60] Default: 1.0

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`check_opt_arg`()	Checks for the optional arguments
`correlation`(r)	Matérn correlation function
`covariance`(r)	Covariance of the model
`default_opt_arg`()	The defaults for the optional arguments:
`default_opt_arg_bounds`()	The defaults boundaries for the optional arguments:
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

check_opt_arg()[source]¶

Checks for the optional arguments

Warns:	nu – If nu is > 50, the model gets numerically unstable.

correlation(r)[source]¶: Matérn correlation function

$\mathrm{cor}(r) = \frac{2^{1-\nu}}{\Gamma\left(\nu\right)} \cdot \left(\sqrt{2\nu}\cdot\frac{r}{\ell}\right)^{\nu} \cdot \mathrm{K}_{\nu}\left(\sqrt{2\nu}\cdot\frac{r}{\ell}\right)$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

default_opt_arg()[source]¶

The defaults for the optional arguments:

{"nu": 1.0}

Returns:	Defaults for optional arguments
Return type:	`dict`

default_opt_arg_bounds()[source]¶

The defaults boundaries for the optional arguments:

{"nu": [0.5, 60.0, "cc"]}

Returns:	Boundaries for optional arguments
Return type:	`dict`

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

class gstools.covmodel.models.MaternRescal(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The rescaled Matérn covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \frac{2^{1-\nu}}{\Gamma\left(\nu\right)} \cdot \left(\frac{\pi}{B\left(\nu,\frac{1}{2}\right)} \cdot \frac{r}{\ell}\right)^{\nu} \cdot \mathrm{K}_{\nu}\left(\frac{\pi}{B\left(\nu,\frac{1}{2}\right)} \cdot \frac{r}{\ell}\right)$

Where $\Gamma$ is the gamma function, $\mathrm{K}_{\nu}$ is the modified Bessel function of the second kind and $B$ is the Euler beta function.

$\nu$ is a shape parameter and should be > 0.5.

Other Parameters:

**opt_arg – The following parameters are covered by these keyword arguments
nu (float, optional) – Shape parameter. Standard range: [0.5, 60] Default: 1.0

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`check_opt_arg`()	Checks for the optional arguments
`correlation`(r)	Rescaled Matérn correlation function
`covariance`(r)	Covariance of the model
`default_opt_arg`()	The defaults for the optional arguments:
`default_opt_arg_bounds`()	The defaults boundaries for the optional arguments:
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

check_opt_arg()[source]¶

Checks for the optional arguments

Warns:	nu – If nu is > 50, the model gets numerically unstable.

correlation(r)[source]¶: Rescaled Matérn correlation function

$\mathrm{cor}(r) = \frac{2^{1-\nu}}{\Gamma\left(\nu\right)} \cdot \left(\frac{\pi}{B\left(\nu,\frac{1}{2}\right)} \cdot \frac{r}{\ell}\right)^{\nu} \cdot \mathrm{K}_{\nu}\left(\frac{\pi}{B\left(\nu,\frac{1}{2}\right)} \cdot\frac{r}{\ell}\right)$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

default_opt_arg()[source]¶

The defaults for the optional arguments:

{"nu": 1.0}

Returns:	Defaults for optional arguments
Return type:	`dict`

default_opt_arg_bounds()[source]¶

The defaults boundaries for the optional arguments:

{"nu": [0.5, 60.0, "cc"]}

Returns:	Boundaries for optional arguments
Return type:	`dict`

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

class gstools.covmodel.models.Linear(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, integral_scale=None, var_raw=None, hankel_kw=None, **opt_arg)[source]¶

Bases: gstools.covmodel.base.CovModel

The bounded linear covariance model

Notes

This model is given by the following correlation function:

$\mathrm{cor}(r) = \begin{cases} 1-\frac{r}{\ell} & r<\ell\\ 0 & r\geq\ell \end{cases}$

Parameters:

dim (int, optional) – dimension of the model. Default: 3
var (float, optional) – variance of the model (the nugget is not included in “this” variance) Default: 1.0
len_scale (float or list, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. Default: 1.0
nugget (float, optional) – nugget of the model. Default: 0.0
anis (float or list, optional) – anisotropy ratios in the transversal directions [y, z]. Default: 1.0
angles (float or list, optional) –
angles of rotation:
in 2D: given as rotation around z-axis

in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default: 0.0
integral_scale (float or list or None, optional) – If given, len_scale will be ignored and recalculated, so that the integral scale of the model matches the given one. Default: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.SymmetricFourierTransform used for the spectrum calculation. Use with caution (Better: Don’t!). None is equivalent to {"a": -1, "b": 1, "N": 1000, "h": 0.001}. Default: None

Methods

`correlation`(r)	Linear correlation function
`covariance`(r)	Covariance of the model
`variogram`(r)	Isotropic variogram of the model
`variogram_normed`(r)	Normalized variogram of the model

correlation(r)[source]¶: Linear correlation function

$\mathrm{cor}(r) = \begin{cases} 1-\frac{r}{\ell} & r<\ell\\ 0 & r\geq\ell \end{cases}$

covariance(r)¶

Covariance of the model

Given by: $C\left(r\right)= \sigma^2\cdot\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram(r)¶

Isotropic variogram of the model

Given by: $\gamma\left(r\right)= \sigma^2\cdot\left(1-\mathrm{cor}\left(r\right)\right)+n$

Where $\mathrm{cor}(r)$ is the correlation function.

variogram_normed(r)¶

Normalized variogram of the model

Given by: $\tilde{\gamma}\left(r\right)= 1-\mathrm{cor}\left(r\right)$

Where $\mathrm{cor}(r)$ is the correlation function.