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53 lines
1.2 KiB
ReStructuredText
53 lines
1.2 KiB
ReStructuredText
.. _api_DataMisfit:
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Data Misfit
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***********
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The data misfit using an l_2 norm is:
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.. math::
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\mu_\text{data} = {1\over 2}\left| \mathbf{W}_d (\mathbf{d}_\text{pred} - \mathbf{d}_\text{obs}) \right|_2^2
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If the field, u, is provided, the calculation of the data is fast:
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.. math::
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\mathbf{d}_\text{pred} = \mathbf{Pu(m)}
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\mathbf{R} = \mathbf{W}_d (\mathbf{d}_\text{pred} - \mathbf{d}_\text{obs})
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Where P is a projection matrix that brings the field on the full domain to the data measurement locations;
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u is the field of interest; d_obs is the observed data; and \\\(\\mathbf{W}_d\\\) is the weighting matrix.
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The derivative of this, with respect to the model, is:
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.. math::
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\frac{\partial \mu_\text{data}}{\partial \mathbf{m}} = \mathbf{J}^\top \mathbf{W}_d \mathbf{R}
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The second derivative is:
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.. math::
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\frac{\partial^2 \mu_\text{data}}{\partial^2 \mathbf{m}} = \mathbf{J}^\top \mathbf{W}_d \mathbf{W}_d \mathbf{J}
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The API
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=======
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.. autoclass:: SimPEG.DataMisfit.BaseDataMisfit
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:members:
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:undoc-members:
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Common Data Misfits
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===================
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l2 norm
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-------
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.. autoclass:: SimPEG.DataMisfit.l2_DataMisfit
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:members:
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:undoc-members:
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