# verde.VectorSpline2D#

class verde.VectorSpline2D(poisson=0.5, mindist=10000.0, damping=None, force_coords=None, engine='auto')[source]#

Elastically coupled interpolation of 2-component vector data.

This gridder assumes Cartesian coordinates.

Uses the Green’s functions based on elastic deformation from [SandwellWessel2016]. The interpolation is done by estimating point forces that generate an elastic deformation that fits the observed vector data. The deformation equations are based on a 2D elastic sheet with a constant Poisson’s ratio. The data can then be predicted at any desired location.

The east and north data components are coupled through the elastic deformation equations. This coupling is controlled by the Poisson’s ratio, which is usually between -1 and 1. The special case of Poisson’s ratio -1 leads to an uncoupled interpolation, meaning that the east and north components don’t interfere with each other.

The point forces are traditionally placed under each data point. The force locations are set the first time `fit` is called. Subsequent calls will fit using the same force locations as the first call. This configuration results in an exact prediction at the data points but can be unstable.

[SandwellWessel2016] stabilize the solution using Singular Value Decomposition but we use ridge regression instead. The regularization can be controlled using the damping argument. Alternatively, you can specify the position of the forces manually using the force_coords argument. Regularization or forces not coinciding with data points will result in a least-squares estimate, not an exact solution. Note that the least-squares solution is required for data weights to have any effect.

Before fitting, the Jacobian (design, sensitivity, feature, etc) matrix for the spline is normalized using `sklearn.preprocessing.StandardScaler` without centering the mean so that the transformation can be undone in the estimated forces.

Parameters:
poisson`float`

The Poisson’s ratio for the elastic deformation Green’s functions. Default is 0.5. A value of -1 will lead to uncoupled interpolation of the east and north data components.

mindist`float`

A minimum distance between the point forces and data points. Needed because the Green’s functions are singular when forces and data points coincide. Acts as a fudge factor. A good rule of thumb is to use the average spacing between data points.

damping

The positive damping regularization parameter. Controls how much smoothness is imposed on the estimated forces. If None, no regularization is used.

force_coords`None` or `tuple` `of` `arrays`

The easting and northing coordinates of the point forces. If None (default), then will be set to the data coordinates the first time `fit` is called.

engine`str`

DEPRECATED: This option is deprecated and will be removed in Verde v2.0.0. The numba engine will be the only option. Computation engine for the Jacobian matrix and prediction. Can be `'auto'`, `'numba'`, or `'numpy'`. If `'auto'`, will use numba if it is installed or numpy otherwise. The numba version is multi-threaded and usually faster, which makes fitting and predicting faster.

Attributes:
force_`array`

The estimated forces that fit the observed data.

region_`tuple`

The boundaries (`[W, E, S, N]`) of the data used to fit the interpolator. Used as the default region for the `grid` and `scatter` methods.

Methods

 `filter`(coordinates, data[, weights]) Filter the data through the gridder and produce residuals. `fit`(coordinates, data[, weights]) Fit the gridder to the given 2-component vector data. Get metadata routing of this object. `get_params`([deep]) Get parameters for this estimator. `grid`([region, shape, spacing, dims, ...]) Interpolate the data onto a regular grid. `jacobian`(coordinates, force_coords[, dtype]) Make the Jacobian matrix for the 2D coupled elastic deformation. `predict`(coordinates) Evaluate the fitted gridder on the given set of points. `profile`(point1, point2, size[, dims, ...]) Interpolate data along a profile between two points. `scatter`([region, size, random_state, dims, ...]) Interpolate values onto a random scatter of points. `score`(coordinates, data[, weights]) Score the gridder predictions against the given data. `set_fit_request`(*[, coordinates, data, weights]) Request metadata passed to the `fit` method. `set_params`(**params) Set the parameters of this estimator. `set_predict_request`(*[, coordinates]) Request metadata passed to the `predict` method. `set_score_request`(*[, coordinates, data, ...]) Request metadata passed to the `score` method.

## Attributes#

VectorSpline2D.data_names_defaults = [('scalars',), ('east_component', 'north_component'), ('east_component', 'north_component', 'vertical_component')]#
VectorSpline2D.dims = ('northing', 'easting')#
VectorSpline2D.extra_coords_name = 'extra_coord'#

## Methods#

VectorSpline2D.filter(coordinates, data, weights=None)#

Filter the data through the gridder and produce residuals.

Calls `fit` on the data, evaluates the residuals (data - predicted data), and returns the coordinates, residuals, and weights.

Not very useful by itself but this interface makes gridders compatible with other processing operations and is used by `verde.Chain` to join them together (for example, so you can fit a spline on the residuals of a trend).

Parameters:
coordinates`tuple` `of` `arrays`

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). For the specific definition of coordinate systems and what these names mean, see the class docstring.

data`array` or `tuple` `of` `arrays`

The data values of each data point. If the data has more than one component, data must be a tuple of arrays (one for each component).

weights`None` or `array` or `tuple` `of` `arrays`

If not None, then the weights assigned to each data point. If more than one data component is provided, you must provide a weights array for each data component (if not None).

Returns:
`coordinates`, `residuals`, `weights`

The coordinates and weights are same as the input. Residuals are the input data minus the predicted data.

VectorSpline2D.fit(coordinates, data, weights=None)[source]#

Fit the gridder to the given 2-component vector data.

The data region is captured and used as default for the `grid` and `scatter` methods.

All input arrays must have the same shape.

Parameters:
coordinates`tuple` `of` `arrays`

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). Only easting and northing will be used, all subsequent coordinates will be ignored.

data`tuple` `of` `array`

A tuple `(east_component, north_component)` of arrays with the vector data values at each point.

weights

If not None, then the weights assigned to each data point. Must be one array per data component. Typically, this should be 1 over the data uncertainty squared.

Returns:
`self`

Returns this estimator instance for chaining operations.

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:
routing`MetadataRequest`

A `MetadataRequest` encapsulating routing information.

VectorSpline2D.get_params(deep=True)#

Get parameters for this estimator.

Parameters:
deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:
params`dict`

Parameter names mapped to their values.

VectorSpline2D.grid(region=None, shape=None, spacing=None, dims=None, data_names=None, projection=None, coordinates=None, **kwargs)#

Interpolate the data onto a regular grid.

The grid can be specified by two methods:

Use the dims and data_names arguments to set custom names for the dimensions and the data field(s) in the output `xarray.Dataset`. Default names will be provided if none are given.

Parameters:
region`list` = [`W`, `E`, `S`, `N`]

The west, east, south, and north boundaries of a given region. Use only if `coordinates` is None.

shape`tuple` = (`n_north`, `n_east`) `or` `None`

The number of points in the South-North and West-East directions, respectively. Use only if `coordinates` is None.

spacing`tuple` = (`s_north`, `s_east`) `or` `None`

The grid spacing in the South-North and West-East directions, respectively. Use only if `coordinates` is None.

dims

The names of the northing and easting data dimensions, respectively, in the output grid. Default is determined from the `dims` attribute of the class. Must be defined in the following order: northing dimension, easting dimension. NOTE: This is an exception to the “easting” then “northing” pattern but is required for compatibility with xarray.

data_names

The name(s) of the data variables in the output grid. Defaults to `'scalars'` for scalar data, `['east_component', 'north_component']` for 2D vector data, and `['east_component', 'north_component', 'vertical_component']` for 3D vector data.

projection

If not None, then should be a callable object `projection(easting, northing) -> (proj_easting, proj_northing)` that takes in easting and northing coordinate arrays and returns projected northing and easting coordinate arrays. This function will be used to project the generated grid coordinates before passing them into `predict`. For example, you can use this to generate a geographic grid from a Cartesian gridder.

coordinates`tuple` `of` `arrays`

Tuple of arrays containing the coordinates of the grid in the following order: (easting, northing, vertical, …). The easting and northing arrays could be 1d or 2d arrays, if they are 2d they must be part of a meshgrid. If coordinates are passed, `region`, `shape`, and `spacing` are ignored.

Returns:
grid`xarray.Dataset`

The interpolated grid. Metadata about the interpolator is written to the `attrs` attribute.

`verde.grid_coordinates`

Generate the coordinate values for the grid.

VectorSpline2D.jacobian(coordinates, force_coords, dtype='float64')[source]#

Make the Jacobian matrix for the 2D coupled elastic deformation.

The Jacobian is segmented into 4 parts, each relating a force component to a data component [SandwellWessel2016]:

```| J_ee  J_ne |*|f_e| = |d_e|
| J_ne  J_nn | |f_n|   |d_n|
```

The forces and data are assumed to be stacked into 1D arrays with the east component on top of the north component.

Parameters:
coordinates`tuple` `of` `arrays`

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). Only easting and northing will be used, all subsequent coordinates will be ignored.

force_coords`tuple` `of` `arrays`

Arrays with the coordinates for the forces. Should be in the same order as the coordinate arrays.

dtype`str` or `numpy` `dtype`

The type of the Jacobian array.

Returns:
jacobian

The (n_data*2, n_forces*2) Jacobian matrix.

VectorSpline2D.predict(coordinates)[source]#

Evaluate the fitted gridder on the given set of points.

Requires a fitted estimator (see `fit`).

Parameters:
coordinates`tuple` `of` `arrays`

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). Only easting and northing will be used, all subsequent coordinates will be ignored.

Returns:
data`tuple` `of` `arrays`

A tuple `(east_component, north_component)` of arrays with the predicted vector data values at each point.

VectorSpline2D.profile(point1, point2, size, dims=None, data_names=None, projection=None, **kwargs)#

Interpolate data along a profile between two points.

Generates the profile along a straight line assuming Cartesian distances. Point coordinates are generated by `verde.profile_coordinates`. Other arguments for this function can be passed as extra keyword arguments (`kwargs`) to this method.

Use the dims and data_names arguments to set custom names for the dimensions and the data field(s) in the output `pandas.DataFrame`. Default names are provided.

Includes the calculated Cartesian distance from point1 for each data point in the profile.

To specify point1 and point2 in a coordinate system that would require projection to Cartesian (geographic longitude and latitude, for example), use the `projection` argument. With this option, the input points will be projected using the given projection function prior to computations. The generated Cartesian profile coordinates will be projected back to the original coordinate system. Note that the profile points are evenly spaced in projected coordinates, not the original system (e.g., geographic).

Warning

The profile calculation method with a projection has changed in Verde 1.4.0. Previous versions generated coordinates (assuming they were Cartesian) and projected them afterwards. This led to “distances” being incorrectly handled and returned in unprojected coordinates. For example, if `projection` is from geographic to Mercator, the distances would be “angles” (incorrectly calculated as if they were Cartesian). After 1.4.0, point1 and point2 are projected prior to generating coordinates for the profile, guaranteeing that distances are properly handled in a Cartesian system. With this change, the profile points are now evenly spaced in projected coordinates and the distances are returned in projected coordinates as well.

Parameters:
point1`tuple`

The easting and northing coordinates, respectively, of the first point.

point2`tuple`

The easting and northing coordinates, respectively, of the second point.

size`int`

The number of points to generate.

dims

The names of the northing and easting data dimensions, respectively, in the output dataframe. Default is determined from the `dims` attribute of the class. Must be defined in the following order: northing dimension, easting dimension. NOTE: This is an exception to the “easting” then “northing” pattern but is required for compatibility with xarray.

data_names

The name(s) of the data variables in the output dataframe. Defaults to `'scalars'` for scalar data, `['east_component', 'north_component']` for 2D vector data, and `['east_component', 'north_component', 'vertical_component']` for 3D vector data.

projection

If not None, then should be a callable object ```projection(easting, northing, inverse=False) -> (proj_easting, proj_northing)``` that takes in easting and northing coordinate arrays and returns projected northing and easting coordinate arrays. Should also take an optional keyword argument `inverse` (default to False) that if True will calculate the inverse transform instead. This function will be used to project the profile end points before generating coordinates and passing them into `predict`. It will also be used to undo the projection of the coordinates before returning the results.

Returns:
table`pandas.DataFrame`

The interpolated values along the profile.

VectorSpline2D.scatter(region=None, size=300, random_state=0, dims=None, data_names=None, projection=None, **kwargs)#

Interpolate values onto a random scatter of points.

Point coordinates are generated by `verde.scatter_points`. Other arguments for this function can be passed as extra keyword arguments (`kwargs`) to this method.

If the interpolator collected the input data region, then it will be used if `region=None`. Otherwise, you must specify the grid region.

Use the dims and data_names arguments to set custom names for the dimensions and the data field(s) in the output `pandas.DataFrame`. Default names are provided.

Warning

The `scatter` method is deprecated and will be removed in Verde 2.0.0. Use `verde.scatter_points` and the `predict` method instead.

Parameters:
region`list` = [`W`, `E`, `S`, `N`]

The west, east, south, and north boundaries of a given region.

size`int`

The number of points to generate.

random_state`numpy.random.RandomState` or `an` `int` `seed`

A random number generator used to define the state of the random permutations. Use a fixed seed to make sure computations are reproducible. Use `None` to choose a seed automatically (resulting in different numbers with each run).

dims

The names of the northing and easting data dimensions, respectively, in the output dataframe. Default is determined from the `dims` attribute of the class. Must be defined in the following order: northing dimension, easting dimension. NOTE: This is an exception to the “easting” then “northing” pattern but is required for compatibility with xarray.

data_names

The name(s) of the data variables in the output dataframe. Defaults to `'scalars'` for scalar data, `['east_component', 'north_component']` for 2D vector data, and `['east_component', 'north_component', 'vertical_component']` for 3D vector data.

projection

If not None, then should be a callable object `projection(easting, northing) -> (proj_easting, proj_northing)` that takes in easting and northing coordinate arrays and returns projected northing and easting coordinate arrays. This function will be used to project the generated scatter coordinates before passing them into `predict`. For example, you can use this to generate a geographic scatter from a Cartesian gridder.

Returns:
table`pandas.DataFrame`

The interpolated values on a random set of points.

VectorSpline2D.score(coordinates, data, weights=None)#

Score the gridder predictions against the given data.

Calculates the R^2 coefficient of determination of between the predicted values and the given data values. A maximum score of 1 means a perfect fit. The score can be negative.

Warning

The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly. The negative version will be used to maintain the behaviour of larger scores being better, which is more compatible with current model selection code.

If the data has more than 1 component, the scores of each component will be averaged.

Parameters:
coordinates`tuple` `of` `arrays`

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). For the specific definition of coordinate systems and what these names mean, see the class docstring.

data`array` or `tuple` `of` `arrays`

The data values of each data point. If the data has more than one component, data must be a tuple of arrays (one for each component).

weights`None` or `array` or `tuple` `of` `arrays`

If not None, then the weights assigned to each data point. If more than one data component is provided, you must provide a weights array for each data component (if not None).

Returns:
score`float`

The R^2 score

VectorSpline2D.set_fit_request(*, coordinates: = '\$UNCHANGED\$', data: = '\$UNCHANGED\$', weights: = '\$UNCHANGED\$') #

Request metadata passed to the `fit` method.

Note that this method is only relevant if `enable_metadata_routing=True` (see `sklearn.set_config`). Please see User Guide on how the routing mechanism works.

The options for each parameter are:

• `True`: metadata is requested, and passed to `fit` if provided. The request is ignored if metadata is not provided.

• `False`: metadata is not requested and the meta-estimator will not pass it to `fit`.

• `None`: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• `str`: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (`sklearn.utils.metadata_routing.UNCHANGED`) retains the existing request. This allows you to change the request for some parameters and not others.

New in version 1.3.

Note

This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a `Pipeline`. Otherwise it has no effect.

Parameters:
coordinates`str`, `True`, `False`, `or` `None`, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for `coordinates` parameter in `fit`.

data`str`, `True`, `False`, `or` `None`, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for `data` parameter in `fit`.

weights`str`, `True`, `False`, `or` `None`, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for `weights` parameter in `fit`.

Returns:
self`object`

The updated object.

VectorSpline2D.set_params(**params)#

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as `Pipeline`). The latter have parameters of the form `<component>__<parameter>` so that it’s possible to update each component of a nested object.

Parameters:
**params`dict`

Estimator parameters.

Returns:
self`estimator` `instance`

Estimator instance.

VectorSpline2D.set_predict_request(*, coordinates: = '\$UNCHANGED\$') #

Request metadata passed to the `predict` method.

Note that this method is only relevant if `enable_metadata_routing=True` (see `sklearn.set_config`). Please see User Guide on how the routing mechanism works.

The options for each parameter are:

• `True`: metadata is requested, and passed to `predict` if provided. The request is ignored if metadata is not provided.

• `False`: metadata is not requested and the meta-estimator will not pass it to `predict`.

• `None`: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• `str`: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (`sklearn.utils.metadata_routing.UNCHANGED`) retains the existing request. This allows you to change the request for some parameters and not others.

New in version 1.3.

Note

This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a `Pipeline`. Otherwise it has no effect.

Parameters:
coordinates`str`, `True`, `False`, `or` `None`, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for `coordinates` parameter in `predict`.

Returns:
self`object`

The updated object.

VectorSpline2D.set_score_request(*, coordinates: = '\$UNCHANGED\$', data: = '\$UNCHANGED\$', weights: = '\$UNCHANGED\$') #

Request metadata passed to the `score` method.

Note that this method is only relevant if `enable_metadata_routing=True` (see `sklearn.set_config`). Please see User Guide on how the routing mechanism works.

The options for each parameter are:

• `True`: metadata is requested, and passed to `score` if provided. The request is ignored if metadata is not provided.

• `False`: metadata is not requested and the meta-estimator will not pass it to `score`.

• `None`: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• `str`: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (`sklearn.utils.metadata_routing.UNCHANGED`) retains the existing request. This allows you to change the request for some parameters and not others.

New in version 1.3.

Note

This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a `Pipeline`. Otherwise it has no effect.

Parameters:
coordinates`str`, `True`, `False`, `or` `None`, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for `coordinates` parameter in `score`.

data`str`, `True`, `False`, `or` `None`, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for `data` parameter in `score`.

weights`str`, `True`, `False`, `or` `None`, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for `weights` parameter in `score`.

Returns:
self`object`

The updated object.

Vector Data

Vector Data