Richter (1935) and Gutenberg and Richter (1942) developed the local magnitude scale using Wood-Anderson seismographs. Richter (1935) defined ``local'' magnitude measured at a certain station as:
Here is the local magnitude estimate; is the logarithm of the maximum trace amplitude (in mm) recorded by a standard Wood-Anderson torsion seismograph;
is the logarithm of a standard event of magnitude zero at the same epicentral distance in km; and is the adjustment for that station. The estimate for an event is then the arithmetic average of the individual estimates from the Wood-Anderson seismographs that recorded the event.
In the past thirty years, the instrumentation with which we measure earthquakes has changed, and we can now process the data digitally. Nonetheless, we would like to continue to assign events with local magnitudes which are consistent with those that have been calculated in Northern and Southern California in the past.
In 2006-2007, we reported the development of a new
, valid in all of California for hypocentral distances from 1 km to 500 km. In this year, agreement was reached on a set of station corrections, and the CISN parameters were validated by comparing values determined using the ``new'' and ``old'' systems.
Comparison of for Northern California (NC) earthquakes recorded in 2000-2003 determined using the new CISN function and with parameters currently used in NC. Uncertainties exist for both measurements and depend mainly on the number of SNCLs used in the calculation. The slope is close to 1 and the intercept close to 0, indicating good agreement.
A set of approximately 100,000 waveforms from 255 candidate earthquakes recorded by 1160 horizontal channels (Station-Network-Channel-Location, or SNCL labels are used to describe each) from the AZ, BK, CI and NC networks was selected. The candidate events were selected from the CISN 2000-2006 seismicity by gridding the state into 50 km square bins. Two events were selected for each bin: the largest 3+ event, and the largest 3+ event in 2006 (or second largest if the largest 2000-2006 event in the bin occurred in 2006) to obtain adequate data USArray stations. The SNCL channels included both broadband and strong motion instruments. The data set was culled to remove events and SNCLs with fewer than 7 data each and to also remove data with event-SNCL distances greater than 500 km. Local magnitude () was calculated from each channel's trace by deconvolving the instrument response and convolving the response of a Wood-Anderson seismograph (Uhrhammer et al, 1996). For each event, differences for all SNCL pairs were calculated, giving a dataset with more than 10 million differential observations. The differential data were inverted, via constrained linear least-squares with two constraints: (1)
= -3; and (2) the sum of the s for selected SNCLs with historical values = the sum of their historical values. This approach was taken to ensure consistency with past magnitudes determined in Northern and Southern California.
The constrained linear least-squares perturbations to the function were found to be very stable and well represented by a sixth order Chebyshev polynomial at hypocentral distances from 8 km to 500 km hypocentral distance. At shorter distances, it is approximated by a line with a slope close to 2. This form was adopted and an algorithm was developed and used in the subsequent inversions for the SNCL adjustments. The s for the broadband SNCLs were then determined and found to be acceptable for the colocated strong motion sensors as well.
Comparison of for Southern California (SC) earthquakes recorded in 2000-2003 determined using the new CISN function and with parameters currently used in SC. Uncertainties exist for both measurements and depend mainly on the number of SNCLs used in the calculation. The slope is close to 1 and the intercept close to 0, indicating good agreement.
To assess the consistency of the CISN with existing s, we performed three comparisons. In Figure 2.77, values taken from the Northern California database are compared against CISN determined for the same events using the same Wood-Anderson amplitudes. In the figure, the uncertainties for both magnitudes are shown; they depend primarily on the number of SNCLs contributing to the determination. As both values have uncertainties, the best fit line is calculated using a bilinear L1 norm. Reassuringly, the slope of the line is close to one and the intercept close to zero, indicating agreement between the two systems. Figure 2.78 makes a similar comparison for Southern California earthquakes. Here again, the slope is one, to within the uncertainty, and the intercept is zero.
In a final validation step, we selected a set of 96 earthquakes recorded by both Northern and Southern California stations during the interval 2000-2006 (Figure 2.79. CISN values determined only using Northern (horizontal axis) or Southern (vertical axis) SNCLs also agree well (slope=1, intercept=0).
CISN for events from 2000-2006 recorded in both Northern and Southern California. The horizontal axis gives for an event calculated using only Northern California SNCLs while the vertical axis gives the from Southern California SNCLs. The greater uncertainty for Northern California results from the lower density of seismic stations.
The new state-wide and station corrections for determining magnitudes will improve reporting for earthquakes in all of California on several counts. First, was being calculated using only a subset of the currently existing broadband stations in both Northern and Southern California, as only they had been calibrated. In the past 10 years, many broadband stations and strong motion stations have been added to the networks. With the additional stations, determination should become much more reliable. Second, until now Northern (Uhrhammer et al, 1996) and Southern California (Kanamori et al, 1999) have been using different functions, with their attendant values for each SNCL. Thus different magnitudes were often determined by Northern or Southern California for an earthquake if it was near the boundary of the reporting regions, or for very large earthquakes in the other region. Figure 2.79 shows that this will no longer be the case. The CISN system is being used in Southern California since February 2008. It will be implemented in Northern California with the transition to the new event processing software.