# Differential Time Data

Like all algorithms for improving the estimates of relative location for a clustered set of seismic events, **mloc** turns absolute arrival time data into differential times, i.e., the time lag between observations of the same phase at the same station for two different events. The Double Difference algorithm (Waldhauser and Ellsworth, 2000) popularized the use of directly measured differential times, using waveform cross-correlation on digitally-recorded data, for input to a multiple event relocation algorithm.

This approach is especially valuable in the context of routine network operations, where it is possible to automate the processing of consistently-available and well-calibrated digital data streams to obtain the necessary measurements. It is less easily applied to the kinds of analyses typically undertaken with **mloc**, where the set of stations providing data is usually very heterogenous and digital waveforms would be either impossible or very difficult to obtain. Nevertheless, differential time datasets are available for some events of interest, and in keeping with the “omniverous” data usage policy of **mloc** it is desirable to be able to use differential time datasets when they can be obtained.

Since data input to **mloc** is event-based, it is not feasible to reference a second event using the standard MNF v1.3 format. Therefore a special format for reading differential time data in **mloc** was designed: MNF v1.5. No conversion code is supplied with **mloc**; the user will need to write one for each distinct source of differential time data.

## Description

Each line of the input file references two events, by the event names assigned with the even command. If an event referenced in a differential time datum is not found in the cluster, a warning is given.

The input datum is “reduced relative arrival time”, the time difference between the two arrivals if they are treated as occurring on the same day. This is converted to dummy arrival times for the associated pair of events. For the template event the theoretical TT is added to the origin time of the input data file. For the target event, the reduced relative arrival time is added to the template event’s dummy arrival time. Dummy arrival times derived from differential time data are only used for cluster vectors, never the hypocentroid.

Phase names for differential time data are not altered from the input file and phase re-identification is disabled. There is no testing for duplicate readings. Differential time data for Lg are permitted, but otherwise only phases that are in the tau-p phase list are processed.

### Uncertainty

Most (but not all) cross-correlation algorithms produce an estimate of the uncertainty of the differential time datum, but this value underestimates the true uncertainty in the context of relocation even if it is “perfect” in itself because we assume a single theoretical travel time model for the entire cluster, on which the dummy arrival times are based. For a cluster of finite extent (many are 50-100 km across) lateral variations in crustal strucure or even over the full raypath lead to additional scatter.

It is generally true that waveform cross-correlation decreases the scatter in differential time measurements over what is possible by differencing raw arrival time data from standard seismic bulletins such as the ISC Bulletin, but not (in my experience) as much as the formal uncertainties that come with some cross-correlation analyses would imply. This conclusion is based on the application of differential time data to a dozen or so clusters, using measurements from researchers using three different methodologies. Through the **mloc** analysis we obtain empirical reading errors (scatter) for the differential time data just as for “regular” arrival time picks. The empirically-observed scatter (even when the relocation is dominated by the differential time data, is nearly always larger than the formal uncertainty from the cross-correlation analysis.