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.
`Stable`([dim, var, len_scale, nugget, anis, …])	The stable covariance model.
`Rational`([dim, var, len_scale, nugget, …])	The rational quadratic covariance model.
`Linear`([dim, var, len_scale, nugget, anis, …])	The bounded linear covariance model.
`Circular`([dim, var, len_scale, nugget, …])	The circular covariance model.
`Spherical`([dim, var, len_scale, nugget, …])	The Spherical covariance model.
`Intersection`([dim, var, len_scale, nugget, …])	The Intersection 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}.

Attributes:

angles: numpy.ndarray: Rotation angles (in rad) of the model.
anis: numpy.ndarray: The anisotropy factors of the model.
arg: list of str: Names of all arguments.
arg_bounds: dict: Bounds for all parameters.
dim: int: The dimension of the model.
dist_func: tuple of callable: pdf, cdf and ppf.
do_rotation: bool: State if a rotation is performed.
hankel_kw: dict: hankel.SymmetricFourierTransform kwargs.
has_cdf: bool: State if a cdf is defined by the user.
has_ppf: bool: State if a ppf is defined by the user.
integral_scale: float: The main integral scale of the model.
integral_scale_vec: numpy.ndarray: The integral scales in each direction.
len_scale: float: The main length scale of the model.
len_scale_bounds: list: Bounds for the lenght scale.
len_scale_vec: numpy.ndarray: The length scales in each direction.
name: str: The name of the CovModel class.
nugget: float: The nugget of the model.
nugget_bounds: list: Bounds for the nugget.
opt_arg: list of str: Names of the optional arguments.
opt_arg_bounds: dict: Bounds for the optional arguments.
pykrige_angle: 2D rotation angle for pykrige.
pykrige_angle_x: 3D rotation angle around x for pykrige.
pykrige_angle_y: 3D rotation angle around y for pykrige.
pykrige_angle_z: 3D rotation angle around z for pykrige.
pykrige_anis: 2D anisotropy ratio for pykrige.
pykrige_anis_y: 3D anisotropy ratio in y direction for pykrige.
pykrige_anis_z: 3D anisotropy ratio in z direction for pykrige.
pykrige_kwargs: Keyword arguments for pykrige routines.
sill: float: The sill of the variogram.
var: float: The variance of the model.
var_bounds: list: Bounds for the variance.
var_raw: float: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Run checks for the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`correlation`(r)	Gaussian correlation function.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Provide default optional arguments by the user.
`default_opt_arg_bounds`()	Provide default boundaries for optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_cdf`(r)	Radial spectral cdf.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectral_rad_ppf`(u)	Radial spectral ppf.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic variogram of the model.

calc_integral_scale()[source]¶: Calculate the integral scale of the isotrope model.

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_density(k)[source]¶

Spectral density of the covariance model.

This is given by:

$\tilde{S}(k) = \frac{S(k)}{\sigma^2}$

Where $S(k)$ is the spectrum of the covariance model.

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

spectral_rad_cdf(r)[source]¶: Radial spectral cdf.

spectral_rad_ppf(u)[source]¶: Radial spectral ppf.

Notes

Not defined for 3D.

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.

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}.

Attributes:

angles: numpy.ndarray: Rotation angles (in rad) of the model.
anis: numpy.ndarray: The anisotropy factors of the model.
arg: list of str: Names of all arguments.
arg_bounds: dict: Bounds for all parameters.
dim: int: The dimension of the model.
dist_func: tuple of callable: pdf, cdf and ppf.
do_rotation: bool: State if a rotation is performed.
hankel_kw: dict: hankel.SymmetricFourierTransform kwargs.
has_cdf: bool: State if a cdf is defined by the user.
has_ppf: bool: State if a ppf is defined by the user.
integral_scale: float: The main integral scale of the model.
integral_scale_vec: numpy.ndarray: The integral scales in each direction.
len_scale: float: The main length scale of the model.
len_scale_bounds: list: Bounds for the lenght scale.
len_scale_vec: numpy.ndarray: The length scales in each direction.
name: str: The name of the CovModel class.
nugget: float: The nugget of the model.
nugget_bounds: list: Bounds for the nugget.
opt_arg: list of str: Names of the optional arguments.
opt_arg_bounds: dict: Bounds for the optional arguments.
pykrige_angle: 2D rotation angle for pykrige.
pykrige_angle_x: 3D rotation angle around x for pykrige.
pykrige_angle_y: 3D rotation angle around y for pykrige.
pykrige_angle_z: 3D rotation angle around z for pykrige.
pykrige_anis: 2D anisotropy ratio for pykrige.
pykrige_anis_y: 3D anisotropy ratio in y direction for pykrige.
pykrige_anis_z: 3D anisotropy ratio in z direction for pykrige.
pykrige_kwargs: Keyword arguments for pykrige routines.
sill: float: The sill of the variogram.
var: float: The variance of the model.
var_bounds: list: Bounds for the variance.
var_raw: float: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Run checks for the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`correlation`(r)	Exponential correlation function.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Provide default optional arguments by the user.
`default_opt_arg_bounds`()	Provide default boundaries for optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_cdf`(r)	Radial spectral cdf.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectral_rad_ppf`(u)	Radial spectral ppf.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic variogram of the model.

calc_integral_scale()[source]¶: Calculate the integral scale of the isotrope model.

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_density(k)[source]¶

Spectral density of the covariance model.

This is given by:

$\tilde{S}(k) = \frac{S(k)}{\sigma^2}$

Where $S(k)$ is the spectrum of the covariance model.

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

spectral_rad_cdf(r)[source]¶: Radial spectral cdf.

spectral_rad_ppf(u)[source]¶: Radial spectral ppf.

Notes

Not defined for 3D.

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.

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:

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

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 >= 0.2.

If $\nu > 20$ , a gaussian model is used, since it is the limit case:

$\mathrm{cor}(r) = \exp\left(- \frac{1}{4} \cdot \left(\frac{r}{\ell}\right)^2\right)$

Other Parameters:
	**opt_arg – The following parameters are covered by these keyword arguments nu (`float`, optional) – Shape parameter. Standard range: `[0.2, 30]` 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}`.
Attributes:	`angles` `numpy.ndarray`: Rotation angles (in rad) of the model. `anis` `numpy.ndarray`: The anisotropy factors of the model. `arg` `list` of `str`: Names of all arguments. `arg_bounds` `dict`: Bounds for all parameters. `dim` `int`: The dimension of the model. `dist_func` `tuple` of `callable`: pdf, cdf and ppf. `do_rotation` `bool`: State if a rotation is performed. `hankel_kw` `dict`: `hankel.SymmetricFourierTransform` kwargs. `has_cdf` `bool`: State if a cdf is defined by the user. `has_ppf` `bool`: State if a ppf is defined by the user. `integral_scale` `float`: The main integral scale of the model. `integral_scale_vec` `numpy.ndarray`: The integral scales in each direction. `len_scale` `float`: The main length scale of the model. `len_scale_bounds` `list`: Bounds for the lenght scale. `len_scale_vec` `numpy.ndarray`: The length scales in each direction. `name` `str`: The name of the CovModel class. `nugget` `float`: The nugget of the model. `nugget_bounds` `list`: Bounds for the nugget. `opt_arg` `list` of `str`: Names of the optional arguments. `opt_arg_bounds` `dict`: Bounds for the optional arguments. `pykrige_angle` 2D rotation angle for pykrige. `pykrige_angle_x` 3D rotation angle around x for pykrige. `pykrige_angle_y` 3D rotation angle around y for pykrige. `pykrige_angle_z` 3D rotation angle around z for pykrige. `pykrige_anis` 2D anisotropy ratio for pykrige. `pykrige_anis_y` 3D anisotropy ratio in y direction for pykrige. `pykrige_anis_z` 3D anisotropy ratio in z direction for pykrige. `pykrige_kwargs` Keyword arguments for pykrige routines. `sill` `float`: The sill of the variogram. `var` `float`: The variance of the model. `var_bounds` `list`: Bounds for the variance. `var_raw` `float`: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Run checks for the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`correlation`(r)	Matérn correlation function.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Defaults for the optional arguments.
`default_opt_arg_bounds`()	Defaults for boundaries of the optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic variogram of the model.

calc_integral_scale()[source]¶: Calculate the integral scale of the isotrope model.

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

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

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]¶

Defaults for the optional arguments.

{"nu": 1.0}

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

default_opt_arg_bounds()[source]¶

Defaults for boundaries of the optional arguments.

{"nu": [0.5, 30.0, "cc"]}

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

spectral_density(k)[source]¶

Spectral density of the covariance model.

This is given by:

$\tilde{S}(k) = \frac{S(k)}{\sigma^2}$

Where $S(k)$ is the spectrum of the covariance 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.

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}`.
Attributes:	`angles` `numpy.ndarray`: Rotation angles (in rad) of the model. `anis` `numpy.ndarray`: The anisotropy factors of the model. `arg` `list` of `str`: Names of all arguments. `arg_bounds` `dict`: Bounds for all parameters. `dim` `int`: The dimension of the model. `dist_func` `tuple` of `callable`: pdf, cdf and ppf. `do_rotation` `bool`: State if a rotation is performed. `hankel_kw` `dict`: `hankel.SymmetricFourierTransform` kwargs. `has_cdf` `bool`: State if a cdf is defined by the user. `has_ppf` `bool`: State if a ppf is defined by the user. `integral_scale` `float`: The main integral scale of the model. `integral_scale_vec` `numpy.ndarray`: The integral scales in each direction. `len_scale` `float`: The main length scale of the model. `len_scale_bounds` `list`: Bounds for the lenght scale. `len_scale_vec` `numpy.ndarray`: The length scales in each direction. `name` `str`: The name of the CovModel class. `nugget` `float`: The nugget of the model. `nugget_bounds` `list`: Bounds for the nugget. `opt_arg` `list` of `str`: Names of the optional arguments. `opt_arg_bounds` `dict`: Bounds for the optional arguments. `pykrige_angle` 2D rotation angle for pykrige. `pykrige_angle_x` 3D rotation angle around x for pykrige. `pykrige_angle_y` 3D rotation angle around y for pykrige. `pykrige_angle_z` 3D rotation angle around z for pykrige. `pykrige_anis` 2D anisotropy ratio for pykrige. `pykrige_anis_y` 3D anisotropy ratio in y direction for pykrige. `pykrige_anis_z` 3D anisotropy ratio in z direction for pykrige. `pykrige_kwargs` Keyword arguments for pykrige routines. `sill` `float`: The sill of the variogram. `var` `float`: The variance of the model. `var_bounds` `list`: Bounds for the variance. `var_raw` `float`: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Check the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`correlation`(r)	Stable correlation function.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Defaults for the optional arguments.
`default_opt_arg_bounds`()	Defaults for boundaries of the optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic variogram of the model.

check_opt_arg()[source]¶

Check 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]¶

Defaults for the optional arguments.

{"alpha": 1.5}

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

default_opt_arg_bounds()[source]¶

Defaults for boundaries of 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.

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}`.
Attributes:	`angles` `numpy.ndarray`: Rotation angles (in rad) of the model. `anis` `numpy.ndarray`: The anisotropy factors of the model. `arg` `list` of `str`: Names of all arguments. `arg_bounds` `dict`: Bounds for all parameters. `dim` `int`: The dimension of the model. `dist_func` `tuple` of `callable`: pdf, cdf and ppf. `do_rotation` `bool`: State if a rotation is performed. `hankel_kw` `dict`: `hankel.SymmetricFourierTransform` kwargs. `has_cdf` `bool`: State if a cdf is defined by the user. `has_ppf` `bool`: State if a ppf is defined by the user. `integral_scale` `float`: The main integral scale of the model. `integral_scale_vec` `numpy.ndarray`: The integral scales in each direction. `len_scale` `float`: The main length scale of the model. `len_scale_bounds` `list`: Bounds for the lenght scale. `len_scale_vec` `numpy.ndarray`: The length scales in each direction. `name` `str`: The name of the CovModel class. `nugget` `float`: The nugget of the model. `nugget_bounds` `list`: Bounds for the nugget. `opt_arg` `list` of `str`: Names of the optional arguments. `opt_arg_bounds` `dict`: Bounds for the optional arguments. `pykrige_angle` 2D rotation angle for pykrige. `pykrige_angle_x` 3D rotation angle around x for pykrige. `pykrige_angle_y` 3D rotation angle around y for pykrige. `pykrige_angle_z` 3D rotation angle around z for pykrige. `pykrige_anis` 2D anisotropy ratio for pykrige. `pykrige_anis_y` 3D anisotropy ratio in y direction for pykrige. `pykrige_anis_z` 3D anisotropy ratio in z direction for pykrige. `pykrige_kwargs` Keyword arguments for pykrige routines. `sill` `float`: The sill of the variogram. `var` `float`: The variance of the model. `var_bounds` `list`: Bounds for the variance. `var_raw` `float`: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Run checks for the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`correlation`(r)	Rational correlation function.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Defaults for the optional arguments.
`default_opt_arg_bounds`()	Defaults for boundaries of the optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic 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]¶

Defaults for the optional arguments.

{"alpha": 1.0}

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

default_opt_arg_bounds()[source]¶

Defaults for boundaries of 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.

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.

This model is derived from the relative intersection area of two lines in 1D, where the middle points have a distance of $r$ and the line lengths are $\ell$ .

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}.

Attributes:

angles: numpy.ndarray: Rotation angles (in rad) of the model.
anis: numpy.ndarray: The anisotropy factors of the model.
arg: list of str: Names of all arguments.
arg_bounds: dict: Bounds for all parameters.
dim: int: The dimension of the model.
dist_func: tuple of callable: pdf, cdf and ppf.
do_rotation: bool: State if a rotation is performed.
hankel_kw: dict: hankel.SymmetricFourierTransform kwargs.
has_cdf: bool: State if a cdf is defined by the user.
has_ppf: bool: State if a ppf is defined by the user.
integral_scale: float: The main integral scale of the model.
integral_scale_vec: numpy.ndarray: The integral scales in each direction.
len_scale: float: The main length scale of the model.
len_scale_bounds: list: Bounds for the lenght scale.
len_scale_vec: numpy.ndarray: The length scales in each direction.
name: str: The name of the CovModel class.
nugget: float: The nugget of the model.
nugget_bounds: list: Bounds for the nugget.
opt_arg: list of str: Names of the optional arguments.
opt_arg_bounds: dict: Bounds for the optional arguments.
pykrige_angle: 2D rotation angle for pykrige.
pykrige_angle_x: 3D rotation angle around x for pykrige.
pykrige_angle_y: 3D rotation angle around y for pykrige.
pykrige_angle_z: 3D rotation angle around z for pykrige.
pykrige_anis: 2D anisotropy ratio for pykrige.
pykrige_anis_y: 3D anisotropy ratio in y direction for pykrige.
pykrige_anis_z: 3D anisotropy ratio in z direction for pykrige.
pykrige_kwargs: Keyword arguments for pykrige routines.
sill: float: The sill of the variogram.
var: float: The variance of the model.
var_bounds: list: Bounds for the variance.
var_raw: float: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Run checks for the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`correlation`(r)	Linear correlation function.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Provide default optional arguments by the user.
`default_opt_arg_bounds`()	Provide default boundaries for optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic 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.

class gstools.covmodel.models.Circular(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 circular covariance model.

This model is derived as the relative intersection area of two discs in 2D, where the middle points have a distance of $r$ and the diameters are given by $\ell$ .

Notes

This model is given by the following correlation function:

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

.

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}.

Attributes:

angles: numpy.ndarray: Rotation angles (in rad) of the model.
anis: numpy.ndarray: The anisotropy factors of the model.
arg: list of str: Names of all arguments.
arg_bounds: dict: Bounds for all parameters.
dim: int: The dimension of the model.
dist_func: tuple of callable: pdf, cdf and ppf.
do_rotation: bool: State if a rotation is performed.
hankel_kw: dict: hankel.SymmetricFourierTransform kwargs.
has_cdf: bool: State if a cdf is defined by the user.
has_ppf: bool: State if a ppf is defined by the user.
integral_scale: float: The main integral scale of the model.
integral_scale_vec: numpy.ndarray: The integral scales in each direction.
len_scale: float: The main length scale of the model.
len_scale_bounds: list: Bounds for the lenght scale.
len_scale_vec: numpy.ndarray: The length scales in each direction.
name: str: The name of the CovModel class.
nugget: float: The nugget of the model.
nugget_bounds: list: Bounds for the nugget.
opt_arg: list of str: Names of the optional arguments.
opt_arg_bounds: dict: Bounds for the optional arguments.
pykrige_angle: 2D rotation angle for pykrige.
pykrige_angle_x: 3D rotation angle around x for pykrige.
pykrige_angle_y: 3D rotation angle around y for pykrige.
pykrige_angle_z: 3D rotation angle around z for pykrige.
pykrige_anis: 2D anisotropy ratio for pykrige.
pykrige_anis_y: 3D anisotropy ratio in y direction for pykrige.
pykrige_anis_z: 3D anisotropy ratio in z direction for pykrige.
pykrige_kwargs: Keyword arguments for pykrige routines.
sill: float: The sill of the variogram.
var: float: The variance of the model.
var_bounds: list: Bounds for the variance.
var_raw: float: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Run checks for the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`correlation`(r)	Circular correlation function.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Provide default optional arguments by the user.
`default_opt_arg_bounds`()	Provide default boundaries for optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic variogram of the model.

correlation(r)[source]¶: Circular correlation function.

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

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.

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.

This model is derived from the relative intersection area of two spheres in 3D, where the middle points have a distance of $r$ and the diameters are given by $\ell$ .

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}.

Attributes:

angles: numpy.ndarray: Rotation angles (in rad) of the model.
anis: numpy.ndarray: The anisotropy factors of the model.
arg: list of str: Names of all arguments.
arg_bounds: dict: Bounds for all parameters.
dim: int: The dimension of the model.
dist_func: tuple of callable: pdf, cdf and ppf.
do_rotation: bool: State if a rotation is performed.
hankel_kw: dict: hankel.SymmetricFourierTransform kwargs.
has_cdf: bool: State if a cdf is defined by the user.
has_ppf: bool: State if a ppf is defined by the user.
integral_scale: float: The main integral scale of the model.
integral_scale_vec: numpy.ndarray: The integral scales in each direction.
len_scale: float: The main length scale of the model.
len_scale_bounds: list: Bounds for the lenght scale.
len_scale_vec: numpy.ndarray: The length scales in each direction.
name: str: The name of the CovModel class.
nugget: float: The nugget of the model.
nugget_bounds: list: Bounds for the nugget.
opt_arg: list of str: Names of the optional arguments.
opt_arg_bounds: dict: Bounds for the optional arguments.
pykrige_angle: 2D rotation angle for pykrige.
pykrige_angle_x: 3D rotation angle around x for pykrige.
pykrige_angle_y: 3D rotation angle around y for pykrige.
pykrige_angle_z: 3D rotation angle around z for pykrige.
pykrige_anis: 2D anisotropy ratio for pykrige.
pykrige_anis_y: 3D anisotropy ratio in y direction for pykrige.
pykrige_anis_z: 3D anisotropy ratio in z direction for pykrige.
pykrige_kwargs: Keyword arguments for pykrige routines.
sill: float: The sill of the variogram.
var: float: The variance of the model.
var_bounds: list: Bounds for the variance.
var_raw: float: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Run checks for the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`correlation`(r)	Spherical correlation function.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Provide default optional arguments by the user.
`default_opt_arg_bounds`()	Provide default boundaries for optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic 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.

class gstools.covmodel.models.Intersection(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 Intersection covariance model.

This model is derived from the relative intersection area of two d-dimensional spheres, where the middle points have a distance of $r$ and the diameters are given by $\ell$ .

In 1D this is the Linear model, in 2D this is the Circular model and in 3D this is the Spherical model.

Notes

This model is given by the following correlation functions.

In 1D:

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

In 2D:

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

In 3D:

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

.

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}.

Attributes:

angles: numpy.ndarray: Rotation angles (in rad) of the model.
anis: numpy.ndarray: The anisotropy factors of the model.
arg: list of str: Names of all arguments.
arg_bounds: dict: Bounds for all parameters.
dim: int: The dimension of the model.
dist_func: tuple of callable: pdf, cdf and ppf.
do_rotation: bool: State if a rotation is performed.
hankel_kw: dict: hankel.SymmetricFourierTransform kwargs.
has_cdf: bool: State if a cdf is defined by the user.
has_ppf: bool: State if a ppf is defined by the user.
integral_scale: float: The main integral scale of the model.
integral_scale_vec: numpy.ndarray: The integral scales in each direction.
len_scale: float: The main length scale of the model.
len_scale_bounds: list: Bounds for the lenght scale.
len_scale_vec: numpy.ndarray: The length scales in each direction.
name: str: The name of the CovModel class.
nugget: float: The nugget of the model.
nugget_bounds: list: Bounds for the nugget.
opt_arg: list of str: Names of the optional arguments.
opt_arg_bounds: dict: Bounds for the optional arguments.
pykrige_angle: 2D rotation angle for pykrige.
pykrige_angle_x: 3D rotation angle around x for pykrige.
pykrige_angle_y: 3D rotation angle around y for pykrige.
pykrige_angle_z: 3D rotation angle around z for pykrige.
pykrige_anis: 2D anisotropy ratio for pykrige.
pykrige_anis_y: 3D anisotropy ratio in y direction for pykrige.
pykrige_anis_z: 3D anisotropy ratio in z direction for pykrige.
pykrige_kwargs: Keyword arguments for pykrige routines.
sill: float: The sill of the variogram.
var: float: The variance of the model.
var_bounds: list: Bounds for the variance.
var_raw: float: The raw variance of the model without factor.

Methods

`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`check_arg_bounds`()	Check arguments to be within the given bounds.
`check_opt_arg`()	Run checks for the optional arguments.
`cor_spatial`(pos)	Spatial correlation respecting anisotropy and rotation.
`cov_nugget`(r)	Covariance of the model respecting the nugget at r=0.
`cov_spatial`(pos)	Spatial covariance respecting anisotropy and rotation.
`covariance`(r)	Covariance of the model.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Provide default optional arguments by the user.
`default_opt_arg_bounds`()	Provide default boundaries for optional arguments.
`fit_variogram`(x_data, y_data[, maxfev])	Fiting the isotropic variogram-model to given data.
`fix_dim`()	Set a fix dimension for the model.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`percentile_scale`([per])	Calculate the percentile scale of the isotrope model.
`plot`([func])	Plot a function of a the CovModel.
`pykrige_vario`([args, r])	Isotropic variogram of the model for pykrige.
`set_arg_bounds`(**kwargs)	Set bounds for the parameters of the model.
`spectral_density`(k)	Spectral density of the covariance model.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	Factor for the variance.
`vario_nugget`(r)	Isotropic variogram of the model respecting the nugget at r=0.
`vario_spatial`(pos)	Spatial variogram respecting anisotropy and rotation.
`variogram`(r)	Isotropic variogram of the model.

correlation

correlation(r)[source]¶

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_density(k)[source]¶

Spectral density of the covariance model.

This is given by:

$\tilde{S}(k) = \frac{S(k)}{\sigma^2}$

Where $S(k)$ is the spectrum of the covariance 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.