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281 lines
9.4 KiB
Plaintext
281 lines
9.4 KiB
Plaintext
1 year ago
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.. _global_fixedhypocenter:
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###############
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FixedHypocenter
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###############
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Locator for re-computing source time with fixed hypocenter
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Description
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===========
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Mining-related events are useful as ground truth events (:cite:t:`bondár-2009a`)
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because the epicentre and depth can be constrained by physical inspection.
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Unless a local seismograph network with accurate timing also locates the event,
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and that information is available, the origin time must be estimated in order
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for the event to be useful as ground truth. Existing location algorithms in
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|scname|, including :ref:`Hypo71 <global_locsat>` and :ref:`LOCSAT <global_locsat>`
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do not allow the determination of origin time given a set of arrivals and a
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fixed hypocentre. There is a need, then, for a method of fixed hypocentre
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origin time determination.
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Objectives of this locator are:
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* Inversion of arrival times of phase picks for source time fixing hypocenter location.
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* Compatibility of the method of fixed-hypocentre origin time determination with
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the practise of the Comprehensive Test Ban Treaty Organization (CTBTO).
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* Adaptation of a procedure which is compatible with the other locators supported by |scname|.
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* Adaptation of a procedure which can reproduce results of legacy locators currently
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in use, such as GENLOC :cite:t:`pavlis-2004` and GRL, a
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grid-based locator developed at the Canadian Hazards Information Service (CHIS).
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The implementation of this locator by :term:`gempa GmbH` was initiated and has received
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initial funding from :cite:t:`nrcan`.
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Methodology
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===========
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Given the measured arrival times :math:`t_i^k` of phase :math:`k` at
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station :math:`i`, most methods of earthquake hypocentre location involve
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minimization of the weighted squared sum of the residuals. That is,
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minimization of:
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.. math::
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|r_w|^2 = \sum_{i=1}^N {w_i^2 [ t_i^k - \tau - T_{model}^k(r_i,x) ]^2}
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The residuals are computed by subtracting the expected arrival times
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:math:`\tau - T_{model}^k(r_i,x)` based on a velocity model applied at the
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coordinates of each station
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:math:`r_i`.
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Typically the weights can be a combination of the inverse of the
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estimated pick uncertainty :math:`1/{\sigma}_i`, a distance term
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:math:`d^k(\Delta)` and/or a residual weight term :math:`p(r_i)`.
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Alternative weighting schemes can be applied but in this
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implementation we weight by pick uncertainty alone: :math:`w_i=\frac{1}{{\sigma}_i}`.
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In the general case, the model is a nonlinear function of its inputs, and there
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is no analytic solution for the origin time and hypocenter that minimize the
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norm. Typically, the solution is found iteratively, based on an initial guess
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for the origin time and hypocenter. This is the normal procedure for an earthquake
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without an a priori estimate of the hypocentral location.
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When the hypocenter is in fact accurately constrained, the modeled travel time
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is a constant, so we can project each phase arrival back to an equivalent origin
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time
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.. math ::
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\tau_i^k = t_i^k - T_{model}^k (r_i,x)
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so that we only have to find which minimizes:
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.. math::
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|r_w|^2 = \sum_{i=1}^{N}w_i^2 [\tau_i^k - \tau]^2
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The residuals are minimized by:
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.. math::
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\tau = \frac{\sum_{i=1}^{N}w_i^2 (\tau_i^k)^2}{\sum_{i=1}^{N}w_i^2}.
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Thus, the origin time is simply the weighted mean of the equivalent origin
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times, according to the velocity model, associated with the arrivals.
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The standard error of this estimate is:
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.. math::
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\sigma = \sqrt{\frac{\sum_{i=1}^{N}w_i^2 [\tau_i^k - \tau]^2}{\sum_{i=1}^{N}w_i^2}}.
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The methodology for estimating error intervals and ellipses recommended for
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standard processing at the CTBTO (:cite:t:`lee-1975`) is that of
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:cite:t:`jordan-1981` and is implemented
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in LOCSAT (:cite:t:`bratt-1988`).
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Uncertainty is represented by a set of points :math:`x_e` around the final estimate
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:math:`x_f` satisfying:
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.. math::
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\kappa_p^2 &= (x_e - x_f)^TC_m(x_e-x_f), \\
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\kappa_p^2 &= Ms^2F_p(M,K+N-M), \\
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s^2 &= \frac{Ks_K^2+|r_w|^2}{K+N-M}
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where:
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* :math:`C_m`: Covariance matrix, corresponding to the final hypocentre estimate :math:`x_f`.
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* :math:`s^2`: Ratio of actual to assumed.
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* :math:`\kappa_p^2`: The “confidence coefficient” at probability :math:`\rho`.
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* :math:`F_p(m,n)`: Fisher-Snedecor quantile function (inverse cumulative F-distribution)
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for and degrees of freedom of numerator and denominator sum of squares,
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respectively, and probability.
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* :math:`p`: Confidence level: the desired probability that the true epicentre should
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fall within the uncertainty bounds.
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* :math:`N`: Sum of all arrival time, azimuth or slowness estimates. Here, only
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arrival times are considered for inversion.
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* :math:`M`: Number of fitted parameters:
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* 3: error ellipsoid
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* 2: error ellipse
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* 1: depth or time error bounds.
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Here, :math:`M = 1` as we only invert for the time.
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* :math:`s_K^2`: A prior estimate of the ratio of actual to assumed data variances; typically set to 1.
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* :math:`K`: Number of degrees of freedom in prior estimate :math:`s_K^2`.
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:math:`K` can be configured by :confval:`FixedHypocenter.degreesOfFreedom`.
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* :math:`r_w`: Vector of weighted residuals.
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Although this formulation is complex it is useful it because allows the analyst to
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balance a priori and a posteriori estimates of the ratio of actual to assumed
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data variances.
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The covariance matrix in the general case is computed from the weighted sensitivity
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matrix :math:`A_w`, the row-weighted matrix of partial derivatives of arrival
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time with respect to the solution coordinates.
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.. math::
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C_m = A^T_wA_w
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However, when origin time is the only coordinate, the partial derivatives with
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respect to origin time are unity, the weighted sensitivity matrix is simply a
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row vector of weights, and the time-time covariance
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:math:`c_{tt}` is simply the sum of the squares of these weights.
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.. math::
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c_{tt} = \sum_{i=1}^{N}w_i^2
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It is recommended that fixed-hypocentre origin time confidence intervals be
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estimated using the method of :cite:t:`jordan-1981` for error ellipsoids,
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that is, that the time error bounds be represented using
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.. math::
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\Delta t_p &= \sqrt{ \frac{\kappa_p^2}{c_{tt}} } \\
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&= \sqrt{ \frac{F_p(1,K+N-1)}{K+N-1} \frac{Ks_K^2 + \sum_{i=1}^{N}w_i^2 [\tau_i^k-\tau]^2}{\sum_{i=1}^{N}w_i^2}}.
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In addition to recording arrival weights and residuals, distances and azimuths,
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and other details of origin quality, the details of a ground-truth-level (GT1)
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fixed-hypocentre origin time estimate are recorded as:
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* origin.time = :math:`\tau`
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* origin.time_errors.uncertainty = :math:`\Delta t_p`
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* origin.time_errors.confidence_level = :math:`100p`
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* origin.quality.standard_error = :math:`\sigma`
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* origin.quality.ground_truth_level = GT1
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For the sake of reproducibility, a comment is added to every new :term:`origin`
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reporting :math:`K`, :math:`s_K` and :math:`\kappa_p`.
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Application
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===========
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#. Configure the parameters in the section *FixedHypocenter* of the global configuration.
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#. When using in :ref:`scolv` the FixedHypocenter locator can be chose right away
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from the available locators.
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.. figure:: media/scolv-fixedhypocenter.png
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:align: center
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:width: 18cm
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scolv Location tab with FixHypocenter selected for relocating.
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#. Configure the module, e.g. :ref:`screloc` or :ref:`scolv`, which is to use FixedHypocenter:
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* set the locator type / interface: "FixedHypocenter"
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* if requested, set the profile as [interface]/[model], e.g.: LOCSAT/iasp91 or libtau/ak135
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#. Run the module with FixedHypocenter
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Origins created by the FixedHypocenter locator can be identified by the methodID
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and the *confidence/description* comment of the origin paramters, e.g.: ::
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<origin publicID="Origin/20200102030459.123456.8222">
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...
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<timeFixed>false</timeFixed>
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<epicenterFixed>true</epicenterFixed>
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<methodID>FixedHypocenter</methodID>
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<earthModelID>iasp91</earthModelID>
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...
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<comment>
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<text>Confidence coefficient: K-weighted ($K$=8, $s_K$=1 s), $\kappa_p$ = 1.6, $n_{eff}$ = 5.0</text>
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<id>confidence/description</id>
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</comment>
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...
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</origin>
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.. _global_fixedhypocenter_configuration:
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Module Configuration
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====================
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.. note::
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**FixedHypocenter.\***
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*Locator parameters: FixedHypocenter*
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.. confval:: FixedHypocenter.profiles
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Default: ``LOCSAT/iasp91,LOCSAT/tab``
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Type: *list:string*
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Defines a list of available travel time tables. Each item
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is a tuple separated by a slash with format \"[interface]\/[model]\".
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Built\-in interfaces are \"LOCSAT\" and \"libtau\".
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Other interfaces might be added via plugins. Please check their
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documentation for the required interface name.
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.. confval:: FixedHypocenter.usePickUncertainties
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Default: ``false``
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Type: *boolean*
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Whether to use pick time uncertainties rather than a fixed
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time error. If true, then the uncertainties are retrieved from
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each individual pick object. If they are not defined, then the
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default pick time uncertainty as defined by defaultTimeError
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will be used instead.
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.. confval:: FixedHypocenter.defaultTimeError
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Default: ``1.0``
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Type: *double*
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Unit: *s*
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The default pick time uncertainty if pick uncertainties are
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not going to be used or if they are absent.
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.. confval:: FixedHypocenter.degreesOfFreedom
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Default: ``8``
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Type: *int*
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Number of degrees of freedom used for error estimate.
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.. confval:: FixedHypocenter.confLevel
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Default: ``0.9``
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Type: *double*
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Confidence level between 0.5 and 1.
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