Module estimate

This platecurie module contains the following functions:

  • bayes_estimate_cell(): Set up pymc model and estimate the parameters of the magnetic layer model using a probabilistic Bayesian inference method.

  • get_bayes_estimates(): Explore the output of sampling the pymc model

  • L2_estimate_cell(): Set up non-linear curve fitting to estimate the parameters of the magnetic layer model using non-linear least-squares from the function scipy.optimize.curve_fit().

  • get_L2_estimates(): Explore the output the non-linear inversion

  • calculate_psd(): Calculate the analytical PSD function.

An internal function is available to calculate predicted power-spectral density data with a theano decorator to be incorporated as pymc variables. These functions are used within Project methods as with plotting functions.

platecurie.estimate.bayes_estimate_cell(k, psd, epsd, fix_beta=None, prior_zt=None, draws=500, tunes=500, cores=4)

Function to estimate the parameters of the magnetic model at a single cell location of the input grids.

Can incorporate 2 (‘A’, and ‘dz’), 3 (either ‘A’, ‘zt’ and ‘dz’ or ‘A’, ‘dz’, and ‘beta’) or 4 (‘A’, ‘zt’, ‘dz’, and ‘beta’) stochastic variables.

Parameters

  • k (ndarray) – 1D array of wavenumbers

  • psd (ndarray) – 1D array of wavelet scalogram (wavelet PSD)

  • epsd (ndarray) – 1D array of error on wavelet scalogram (wavelet PSD)

  • fix_beta (float, optional) – Fixed beta parameter or estimate it if None

  • prior_zt (float, optional) – Fixed zt parameter or estimate it if None

  • draws (int) – Number of samples to draw from the posterior

  • tunes (int) – Number of samples to discard (burn-in)

  • cores (int) – Number of computer cores to use in calculating independent markov chains

Returns

(tuple): Tuple containing:
  • traceMultiTrace

    Posterior samples from the MCMC chains

  • summaryDataFrame

    Summary statistics from Posterior distributions

  • map_estimatedict

    Container for Maximum a Posteriori (MAP) estimates

platecurie.estimate.get_bayes_estimates(summary, map_estimate)

Extract useful estimates from the Posterior distributions.

Parameters

  • summary (DataFrame) – Summary statistics from Posterior distributions

  • map_estimate (dict) – Container for Maximum a Posteriori (MAP) estimates

Returns

(tuple): tuple containing:

  • mean_A (float) : Mean value of the constant term A

  • std_A (float) : Standard deviation of the constant term A

  • best_A (float) : Most likely value of the constant term A

  • mean_dz (float) : Mean value of the magnetic layer thickness dz

  • std_dz (float) : Standard deviation of magnetic layer thickness dz

  • best_dz (float) : Most likely magnetic layer thickness dz

  • mean_zt (float, optional) : Mean value of the depth to top of layer zt

  • std_zt (float, optional) : Standard deviation of the depth to top of layer zt

  • best_zt (float, optional) : Most likely value of the depth to top of layer zt

  • mean_beta (float, optional) : Mean value of the fractal parameter beta

  • std_beta (float, optional) : Standard deviation of fractal parameter beta

  • best_beat (float, optional) : Most likely fractal parameter beta

platecurie.estimate.L2_estimate_cell(k, psd, epsd, fix_beta=None, prior_zt=None)

Function to estimate the parameters of the magnetic model at a single cell location of the input grids.

Parameters

  • k (ndarray) – 1D array of wavenumbers

  • psd (ndarray) – 1D array of wavelet scalogram (wavelet PSD)

  • epsd (ndarray) – 1D array of error on wavelet scalogram (wavelet PSD)

  • fix_beta (float, optional) – Fixed beta parameter or estimate it if None

  • prior_zt (float, optional) – Fixed zt parameter or estimate it if None

Returns

(tuple): Tuple containing:
  • summaryDataFrame

    Summary statistics from L2 estimation

platecurie.estimate.get_L2_estimates(summary)

Returns digestible estimates from the L2 estimates.

Parameters

summary (DataFrame) – Summary statistics from Posterior distributions

Returns

(tuple): tuple containing:

  • mean_A (float) : Mean value of the constant term A

  • std_A (float) : Standard deviation of the constant term A

  • mean_dz (float) : Mean value of the magnetic layer thickness dz

  • std_dz (float) : Standard deviation of magnetic layer thickness dz

  • mean_zt (float, optional) : Mean value of the depth to top of layer zt

  • std_zt (float, optional) : Standard deviation of the depth to top of layer zt

  • mean_beta (float, optional) : Mean value of the fractal parameter beta

  • std_beta (float, optional) : Standard deviation of fractal parameter beta

  • rchi2 (float) : Reduced chi-squared value

platecurie.estimate.calculate_psd(k, A, zt, dz, beta)

Calculate analytical expression for the power-spectral density function.

Parameters

  • k (ndarray) – 1D array of wavenumbers

  • A (float) – Constant (i.e., DC) value for PSD [1., 30.]

  • zt (float) – Depth to top of magnetized layer (km) [0., 10.]

  • dz (float) – Thickness of magnetized layer (km) [1., 50.]

  • beta (float) – Power-law exponent for fractal magnetization

Returns

(ndarray): Power-spectral density function (shape len(k))

Rubric

References

Audet, P., and Gosselin, J.M. (2019). Curie depth estimation from magnetic anomaly data: a re-assessment using multitaper spectral analysis and Bayesian inference. Geophysical Journal International, 218, 494-507. https://doi.org/10.1093/gji/ggz166

Blakely, R.J., 1995. Potential Theory in Gravity and Magnetic Applications, Cambridge Univ. Press.

platecurie.estimate.calculate_bouligand(k, A, zt, dz, beta)

Calculate the synthetic power spectral density of magnetic anomalies Equation (4) of Bouligand et al. (2009)

Parameters

  • k (ndarray) – 1D array of wavenumbers

  • A (float) – Constant (i.e., DC) value for PSD [1., 30.]

  • zt (float) – Depth to top of magnetized layer (km) [0., 10.]

  • dz (float) – Thickness of magnetized layer (km) [1., 50.]

  • beta (float) – Power-law exponent for fractal magnetization

Returns

(ndarray): Power-spectral density function (shape len(k))

Rubric

References

Bouligand, C., J. M. G. Glen, and R. J. Blakely (2009), Mapping Curie temperature depth in the western United States with a fractal model for crustal magnetization, J. Geophys. Res., 114, B11104, doi:10.1029/2009JB006494