MLOC Depth Constraint

Depth Constraint

Focal depth is a notoriously difficult parameter to resolve in any location algorithm because of the strong trade-off in most cases between focal depth and origin time. mloc has a number of tools with which to extract information on focal depth, subject to whatever limits the dataset imposes. There are three general cases:

  • Focal depth can be resolved as a free parameter.
  • Focal depth must be fixed, but a useful estimate can be made.
  • No information on focal depth is available.

The most relevant commands in mloc are those used to specify which hypocentral parameters will be free in the inversion (freh and frec) and those that establish the starting depth for relocation (the dep_ family).

Free Depth

Focal depth can be included as a free parameter in the relocation of an earthquake when there is arrival time data at a range comparable to several times the focal depth or less. Schematically, what is required is at least one up-going raypath, which will have a partial derivative opposite in sign from that of the down-going raypaths that typically dominate arrival time datasets.

In order to set focal depth as a free parameter for a cluster, every event in the cluster should have suitable data to constrain its depth; otherwise the inversion is unlikely to converge. It is possible, using the frec command, to set depth free for some events and keep it fixed for others, but this has never been tested and it may violate some assumptions of the code. A better strategy is to evaluate the subset of events with depth control separately in a free depth relocation and then set the depths of those events accordingly (depm command) in a fixed-depth relocation of the entire cluster.

Fixed Depth, with Constraint

This is the most common scenario in practice. In the command file for a run each event definition section will include a member of the dep_ family that specifies the depth, its uncertainty and the source of the constraint. The different approaches to determining the depth at which an event should be constrained are discussed below.

No Constraint

In many clusters it is possible to constrain focal depth for at least a few events, and in most source regions it is reasonable to expect most events to occur within a limited depth range. This can be confirmed by reference to the events that do have depth constraint (the depth histogram summary plot helps), and then an average depth or “cluster default depth” can be determined. This value can then be given with the depc command in future command files; it will be applied to any event that does not have its own dep_ command. In most continental regions a default cluster depth of 10-15 km will be reasonable.

Depth Reference Surface

In the global average travel-time model ak135 the zero level for depth is the reference ellipsoid WGS84. In other words, the zero depth reference surface is some distance below the land surface in most continental regions. Like all global location codes mloc normally makes a correction for station elevation (command corr), so the situation is straight-forward for a typical location analysis that uses regional or teleseismic data to estimate the hypocentroid of an event that is deep enough to be below the reference ellipsoid.

The situation gets more complicated if a direct calibration relocation is being done with events that are shallow enough that they may lie above the reference ellipsoid (for a cluster in the Tibet Plateau, for example, the events wouldn’t need to be all that shallow). In this case the corr command can be used to turn off station elevation corrections, which makes the reference surface for depth, roughly, the average elevation of the source region.

mloc is not suited to situations, such as volcano monitoring, where an event may be at a higher elevation than the stations being used to locate it.

Starting Depth

The value of starting depth for a relocation in mloc can be taken from several sources. The order of precedence (from most to least preferred) is:

  • A dep_ command in the command file.
  • Depth read from a .hdf file from a previous run.
  • The depth from the input data file (if it carries a flag as being constrained).
  • The value given with the depc command, if it has been given.
  • The depth from the input data file (if a depth has been provided).

Methods of Constraining Focal Depth

If the available arrival time data are inadequate to resolve focal depth as a free parameter for all events in a cluster the relocation must be done with depths fixed, but it is still possible to determine focal depths for many events with a considerable degree of confidence. There are four methods used in mloc that are based on the available arrival time data:

There are also “external” sources of information on depth that can be used for constraint and mloc includes depth codes for most of the possibilities.

Free-Depth Relocation of a Subset of Events

If some events in a cluster are notable in having readings (say, from temporary stations) at very close range (within several focal depths) it is very worthwhile to form a subcluster for relocation in which depth can be left as a free parameter. In addition to providing very useful information on the likely depth range of the remainder of the cluster, the subcluster can be used to refine a local crustal model which can then be used in the “local-distance” method of depth constraint. The thickness of the crust of the crustal model cannot be reliably constrained unless at least some events have reasonable depth constraint. The depths from the free-depth relocation can be inserted into a subsequent command file as depm constraints in a fixed depth relocation of the larger cluster. mloc includes a command subc that helps in extracting the events that are most likely to behave well in such a subcluster.

Near-Source Readings

The most powerful method of constraining focal depth utilizes readings at epicentral distances that are not much greater than the focal depth. Distances of 2 to 3 times the focal depth are still usable. There must be data (at greater distances, but only crustal arrivals) that constrains the epicenter closely for this to work; the near-source stations will not in themselves be sufficient. Adjustment of focal depth and the crustal model proceed simultaneously. Arrivals at very close range are sensitive to depth more than velocity but the slope of arrival times of more distant crustal arrivals will be sensitive to crustal velocity. When crustal arrivals (out to the Pg/Pn crossover, say, 100-150 km) are well-matched, the thickness of the crust are adjusted to correctly predict the arrival times of Pn and Sn.

In many cases, manual adjustment of the focal depth in this manner is a more accurate method of constraining focal depth than a free-depth relocation, because it emphasizes the fit to the closest stations that are the most sensitive indicator of depth. In a free-depth location all the data used contributes to the estimation of depth, but residuals at greater distances are more likely to be distorted by lateral heterogeneity (or simply a less-accurate assumed crustal velocity model) which can be mapped into a biased estimate of depth. This is evident when a free-depth location leaves the closest stations with significant residuals.

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