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docs: working on method

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Sebastian Heimann 2 years ago
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docs/source/method/index.rst View File

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This document gives an comprehensive overview over Grond's optimisation strategy.
.. contents :: Content
:depth: 3
******
******
Method
******
TODO: REVIEW: This section should be as self-contained as possible, describe the method in general - give references to other sections how things are implemented in Grond.
The very core of any optimisation is the evaluation of a misfit value between observed :math:`{\bf d}_{obs}` and predicted data :math:`{\bf d}_{synth}`. This is most often based on the difference :math:`{\bf d}_{obs} - {\bf d}_{synth}`, but can also be any other comparison, like a correlation measure for example.
`Observed data` here means post-processed data and not the `raw` measurements. E.g. full waveforms are usually tapered to the defined phases, restituted and filtered. `Synthetic waveforms` are the forward- modelled waveforms that are tapered and filtered in the same way as the observed waveforms. Find details on the post-processing in the `targets config`_ section. The `targets` are derived from data defined in the `dataset config`_.
This document gives a comprehensive overview over Grond's methodical background. It describes how the objective function and data weighting are defined, how the optimisation algorithm works and how model uncertainties are
estimated.
This document describes the method of Grond on:
The very core of any optimisation is the evaluation of an objective function or misfit value between observed :math:`{\bf d}_{obs}` and predicted data :math:`{\bf d}_{synth}`. This is most often based on the difference :math:`{\bf d}_{obs} - {\bf d}_{synth}`, but can also be any other comparison, like a correlation measure for example.
1. How Grond implements the differences between :math:`{\bf d}_{obs}` and :math:`{\bf d}_{synth}` with respect to the definition of objective functions and data weighting,
2. how the optimisation is set up to search the model space to find the optimum models and
3. which methods are used to estimate model uncertainties.
`Observed data` here means post-processed data (or features) derived from the `raw` measurements. For example, in the context of seismic source inversion, seismic waveform recordings are usually tapered to extract specific seismic phases, restituted to displacement and filtered. `Predicted data` are in this case forward modelled seismograms that are tapered and filtered in the same way as the observed waveforms.
.. contents :: Content
:depth: 3
Forward modelling with pre-calculated Green's functions
=======================================================
@ -65,9 +55,10 @@ The objective function defines what a `model fit` is and how `good` or `poor` mo
Misfit calculation and objective function
-----------------------------------------
The core of an optimisation is the data-point-wise calculation of the difference between observed and predicted data:
TODO: avoid repetiion
.. math ::
|{\bf d}_{obs} - {\bf d}_{synth}|.
@ -85,6 +76,8 @@ Grond supports different seismological observations and a combination of those,
* from pixel offsets
* measured by using GNSS sensors
TODO: Define the word target.
The misfit is based on the configurable :math:`L^x`-norm with :math:`x \,\, \epsilon \,\, [1, 2, 3, ...]`:
.. math::


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