# 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.

Warning

The VectorSpline2D class is deprecated and will be removed in Verde v2.0.0. Its usage is restricted to GPS/GNSS data and not in the general scope of Verde. Please use the implementation in the Erizo package instead.

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 (None or float) – 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) – Computation engine for the Jacobian matrix and predictions. 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.

Variables
• 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 Summary

 VectorSpline2D.filter(coordinates, data[, …]) Filter the data through the gridder and produce residuals. VectorSpline2D.fit(coordinates, data[, weights]) Fit the gridder to the given 2-component vector data. Get parameters for this estimator. VectorSpline2D.grid([region, shape, …]) Interpolate the data onto a regular grid. VectorSpline2D.jacobian(coordinates, …[, …]) Make the Jacobian matrix for the 2D coupled elastic deformation. VectorSpline2D.predict(coordinates) Evaluate the fitted gridder on the given set of points. VectorSpline2D.profile(point1, point2, size) Interpolate data along a profile between two points. VectorSpline2D.scatter([region, size, …]) Interpolate values onto a random scatter of points. VectorSpline2D.score(coordinates, data[, …]) Score the gridder predictions against the given data. VectorSpline2D.set_params(**params) Set the parameters of this estimator.

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.

No 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 (None or tuple array) – 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.

VectorSpline2D.get_params(deep=True)

Get parameters for this estimator.

Parameters

deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns

params (mapping of string to any) – Parameter names mapped to their values.

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

Interpolate the data onto a regular grid.

The grid can be specified by either the number of points in each dimension (the shape) or by the grid node spacing. See verde.grid_coordinates for details. Other arguments for verde.grid_coordinates 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 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.

• shape (tuple = (n_north, n_east) or None) – The number of points in the South-North and West-East directions, respectively.

• spacing (tuple = (s_north, s_east) or None) – The grid spacing in the South-North and West-East directions, respectively.

• dims (list or None) – 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 (list of None) – 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 (callable or None) – 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.

Returns

grid (xarray.Dataset) – The interpolated grid. Metadata about the interpolator is written to the attrs attribute.

See also

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 (2D array) – 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 (list or None) – 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 (list of None) – 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 (callable or None) – 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.

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 (list or None) – 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 (list of None) – 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 (callable or None) – 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.

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_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 (object) – Estimator instance.