# harmonica.EulerDeconvolution#

class harmonica.EulerDeconvolution(structural_index)[source]#

Estimate source location and base level using Euler Deconvolution

Implements Euler Deconvolution [Reid1990] to estimate subsurface source location and a base level constant from potential field data and their directional derivatives. The approach employs linear least-squares to solve Euler’s homogeneity equation. Assumes a single data window and provides a single estimate.

Hint

Please read the paper [Reid2014] to avoid doing horrible things with Euler deconvolution. [Uieda2014] offer a practical tutorial using legacy Fatiando a Terra code to show some common misinterpretations.

Note

Does not yet support structural index 0.

Parameters:
structural_indexint

Defines the nature of the source of the potential field data. It’s the degree of the field’s rate of change with distance from the source, influencing the decay rate of the field and the formulation of Euler’s homogeneity equation. Correlated with the depth estimate, so larger structural index will lead to larger depths. Choose based on known source geometry. See table below.

Notes

Works on any potential field that satisfies Euler’s homogeneity equation (like gravity, magnetic, and their gradients caused by simple sources):

$(e_i - e_0)\dfrac{\partial f_i}{\partial e} + (n_i - n_0)\dfrac{\partial f_i}{\partial n} + (u_i - u_0)\dfrac{\partial f_i}{\partial u} = \eta (b - f_i),$

in which $$f_i$$ is the given potential field observation at point $$(e_i, n_i, u_i)$$, $$b$$ is the base level (a constant shift of the field, like a regional field), $$\eta$$ is the structural index, and $$(e_0, n_0, u_0)$$ are the coordinates of a point on the source (for a sphere, this is the center point).

The Euler deconvolution estimates $$(e_0, n_0, u_0)$$ and $$b$$ given a potential field and its easting, northing, and upward derivatives and the structural index. However, this assumes that the sources are ideal (see the table below). We recommend reading [ReidThurston2014] for a discussion on what the structural index means and what it does not mean.

After [ReidThurston2014], values of the structural index (SI) can be:

Source type

SI (Mag)

SI (Grav)

Point, sphere

3

2

Line, cylinder, thin bed fault

2

1

Thin sheet edge, thin sill, thin dyke

1

0

Attributes:
location_1d-array

Estimated (easting, northing, upward) coordinates of the source after model fitting.

base_level_float

Estimated base level constant of the anomaly after model fitting.

covariance_2d-array

The 4 x 4 estimated covariance matrix of the solution. Parameters are in the order: easting, northing, upward, base level. This is not an uncertainty of the position but a rough estimate of their variance with regard to the data.

Methods

 fit(coordinates, data) Fit the model using potential field measurements and their derivatives.
EulerDeconvolution.fit(coordinates, data)[source]#

Fit the model using potential field measurements and their derivatives.

Solves Euler’s homogeneity equation to estimate the source location and base level by utilizing field values and their spatial derivatives in easting, northing, and upward directions.

Tip

Data does not need to be gridded for this to work.

Parameters:
coordinatestuple of arrays

Tuple of 3 with the coordinates of each data point. Should be in the following order: (easting, northing, upward). Arrays can be n-dimensional but must all have the same shape.

datatuple of arrays

Tuple of 4 arrays with the observed data in the following order: (potential_field, derivative_easting, derivative_northing, derivative_upward). Arrays can be n-dimensional but must all have the same shape as the coordinates. Derivatives must be in data units over coordinates units, for example nT/m or mGal/m.

Returns:
self

The instance itself, updated with the estimated location_ and base_level_.