35. ML inversion, MAG2

MAG2 is a program to invert for the local magnitude scale ML. The difference to the inversion done in MAG is that MAG2 inverts amplitudes from all events simultaneously for the scale and station corrections. The program can invert for different scale parameters depending on selected distance ranges. The reason for this is that it is known that the geometrical spreading is not the same for example between Pg and Pn. Some authors have suggested distance dependent scales, but most commonly a single scale is used for all distances for simplicity.

The general ML scale is given by

$$ML = log_{10} A + a log_{10}(R) + b R + S + c$$

where we measure the displacement amplitude A in nm, R is the hypocentral distance in km, S is the station correction of the individual stations, and c is a constant added to make the scale comparable to other places at a reference distance. The station corrections add up to 0. The region dependant parameters in the scale are a, accounting for geometrical spreading, and b, accounting for attenuation. The part $(a log_{10}(R) + b R + c)$ is commonly written as $(-log_{10} A0)$.

The program applies singular value decomposition using the Numerical Recipe [Press et al., 2003] routines to invert the observations for a, b and S. It then computes the parameter c based on the reference given through distance, amplitude and magnitude. This allows to calibrate scales between different regions so that they are the same at the reference distance. Commonly c is set such that 480 nm amplitude at 17 km gives ML=2 (this is equivalent to 1 mm on a Wood-Anderson seismograph giving ML=2 at 17 km [Hutton and Boore, 1987]. The original definition was 480 nm at 100 km giving a ML=3 (equivalent to 1 mm on a Wood-Anderson seismograph at 100 km distance), however it is now considred that a shorter reference distance will give a more accurate scale. The inversion can be setup to invert for the geometrical spreading term a in the scale for up to three distance ranges. However, a single attenuation term b is used.

As input the program requires a parameter file mag2.par (or other name can be given as input) in the working directory, and a standard station file e.g. STATION0.HYP. Then the user only has to enter the input file of events in Nordic format, and if needed the name of the parameter file if different from mag2.par. A sample file, mag2.par, and an input file, mag2nor.cat, with events from Norway are in given in DAT.

The parameter file has the following settings given by keywords (any order):

INVERSION TYPE (f10.1) - 1. = singular value decomposition (no other choice yet)
DISTANCES (2f10.1) - distance range in km for observations to use
MINIMUM NUMBER OF OBS/EVEN (f10.1) - only events with more or equal number of observations are used
MIN DISTANCERANGE RATIO (f10.1) - minimum range required computed as ratio of distances defined by DISTANCES
ORIENTATION (f10.1) - use of components: 0. = horizontal and vertical, 1. = horizontal only, 2. = vertical only
SYNTHETIC (f10.1) - set to 1. for synthetic test, scale defined by FIX SCALE A and FIX SCALE B; 0. for inversion of data
NOISE (f10.1) - ratio of amplitude to be added as noise to synthetic test
FIX SCALE A (3f10.1) - set the fixed parameter a in scale, possible for the three distance ranges given by SCALE DISTANCE
FIX SCALE B (3f10.1) - set the fixed b parameter in scale, possible for the three distance ranges given by SCALE DISTANCE
FIX SITE (f10.1) - set to 1. to not invert for station corrections; 0. for default inversion for station corrections
IGNORE COMP (a4) - give component not to be used
IGNORE STAT (a5) - give station not to be used
REFERENCE DISTANCE, REFERENCE AMPLITUDE, REFERENCE MAGNITUDE (all f10.1) - setup of the reference, used to calculate parameter c, give amplitude as Wood Anderson amplitude in mm
SCALE DISTANCE (2f10.1) - give up to two distances which give the transition between the possibly three distance dependent scales; blank or numbers larger than maximum distance will give only one scale
RANGE A (3f10) - The parameter 'a' in the ML scale can be fixed to values given here. Three numbers have to be given, the first gives the start value, the second the end value and the third gives the spacing (e.g., .5,1.5,.1 means that mag2 inverts for 'a' values between .5 and 1.5 with .1 spacing); if this parameter is set, mag2 does run the invesion for the given (fixed) values of 'a'.

The program produces a number of output files:

mag2_amp_dis.out - amplitude versus distance for each event
mag2_amp_obs.out - list of observed amplitudes
mag2_events_read.out - listing of events that were read in
mag2_events_used.out - events used in Nordic format
mag2_evxy.out - coordinates of events used
mag2.out - general output file, lists data used and computed scale
mag2_paths.out - event-station path coordinates
mag2_station_hyp.out - hyp station file with scale and station corrections
mag2_stat_list.out - simple output of stations used
mag2_statxy.out - coordinates of stations used

The output file mag2.out will give some details on the input data, as number of stations, events and observations. It reports the reference used to fix the scale at the reference distance. Next it gives the scale, consisting of three parts if inversion is done for all possible segments. This will be given by a1, a2, a3, while b will be assumed to be the same. First the scale is presented to include the reference distance, second it is shown without the reference distance included with the scale. Then comes a section with the stations and the respective site terms, and finally the list of events with the source term inverted for.

The output file mag2.out for the example in the DAT directory should look like this:

 ML inversion output

 SVD inversion

 Total number of events:           69
 Total number of stations:           23
 Total number of observations:          600

 Reference distance  =    100.0000    
 Reference amplitude =    1.000000    
 Reference magnitude =    3.000000    

 Ml = log A + a log(dist/refdist) + b (dist-refdist) + c + S 
     a1=  0.84717 +/-  0.39844
     a2=  0.00000 +/-  0.00100
     a3=  0.00000 +/-  0.00100
     b =  0.00061 +/-  0.00136
     c =  0.31807 +/-  0.00000

 Ml = log A + a log(dist) + b (dist) + c + S 
    a1 =  0.84717 +/-  0.39844
    a2 =  0.00000 +/-  0.00100
    a3 =  0.00000 +/-  0.00100
     b =  0.00061 +/-  0.00136
    c1 = -1.43679
    c2 =  0.25756
    c3 =  0.25756

Station #    1 STAV  -0.120  +/- 0.2612   58.935    5.702
Station #    2 BLS5  -0.044  +/- 0.2204   59.423    6.456
Station #    3 ODD1  -0.085  +/- 0.2125   59.912    6.628
...
 Average site term:   0.00
  Event #     1 2002051922484590  ML =  1.95 +/- 0.438
  Event #     2 2002052614481700  ML =  1.68 +/- 0.577
  Event #     3 2002063023341400  ML =  1.86 +/- 0.491
...