# Chapter 3: Initial results from the Anza data

## 1. Introduction

The ANZA seismic network was designed to collect seismic data from earthquakes in the neighborhood of the Anza seismic gap [*gThatcher et al.*g, 1975] (see also Chapter I).
These data are recorded with a dynamic range and frequency response which was unprecedented for a permanent seismic network as described in Chapter II. The original joint research
proposals by the USGS and IGPP/UCSD groups focused on the following three principal objectives:

- To determine if there are premonitory changes in seismic observables preceding small and moderate earthquakes and if there are any characteristics of foreshock activity which distinguishes it from the normal seismic background. Analyses will include detailed correlations between earthquake source spectra and other seismological data including space-time seismicity patterns, b-values, fault plane solutions and crustal structure.
- To increase our fundamental knowledge of the mechanisms and seismic spectra of small and moderate earthquakes - particularly for frequencies above the corner frequency, in order to model the earthquake rupture more realistically and to predict strong ground motion.
- To study the relationship between micro-earthquake activity and crustal deformation observed both continuously at PFO and over a wider area by geodetic means.

Fundamental to obtaining these objectives is the precise determination of various parameters which are used to quantify earthquakes. These parameters include the location, magnitude, seismic moment, and stress drop. The estimation of these parameters can be affected by the properties of the earth's crust through which the seismic waves propagate.

Once the database of earthquakes has enough entries, and the effects of the velocity and attenuation structure are understood, detailed studies can be made on groups of events which have differentiable characteristics. During the time that data have been collected with the ANZA network, the analysis has concentrated on the second objective since results could be obtained from small numbers of events. Only now, after six years of data collection, can investigation on the first and third objectives be effectively initiated.

This chapter will present the current state of results and some implications of these results from data collected on the ANZA seismic network. An earlier presentation of results from data recorded during
the first three years of network operation can be found in *gFletcher et al*g. [1987]. This chapter will augment some of the findings of the previous work. The earthquake location procedure will be examined
in section 2. Section 3 will discuss the spatial distribution of events recorded by the network. Section 4 gives the equations used for the source parameter
estimation. Section 5 will discuss the source parameters estimated for these earthquakes. Recent data from borehole seismometers and the effects of borehole vs. surface measurement will
be in section 6. The implications of the stress drop *gvs*g seismic moment results are presented in section 7, followed by the conclusions in section 8.

## 2. Hypocenter Estimation

The first step in the analysis of earthquakes, before any other parameters can be calculated, is to determine accurate locations. The P and S body wave arrivals are picked by Linda Haar at USGS during
their initial processing of the data. Preliminary estimates of the hypocenters for events located within the study area of the ANZA network are calculated using the program HYPOINVERSE [*gKlein*g, 1978].
The velocity model used by this program is the crustal structure of *gHartzell and Brune*g [1979]. These locations have estimated errors of less than 1.5 km for the epicenters and less than 3 km for
the depths [*gFletcher et al.*g, 1987].

These locations do not have the desired accuracy to conduct the studies of the spatial and temporal correlations of earthquakes proposed in the research objectives. There were two ways to improve these
earthquake locations, increase the number of stations used to pick phases, and improve the velocity model. There are twelve stations of the southern California seismic network in the vicinity of the ANZA
network for which body wave arrivals are picked. Jennifer Scott and Guy Masters at IGPP have merged the two sets of arrival times to perform a simultaneous earthquake location and velocity inversion using
the method of *gPavlis and Booker*g [1980] and *Spencer and Gubbins* [1980]. This inversion used 4472 P wave and 2135 S wave arrival times with high quality picks from 22 stations combined from
both networks. Most of the S wave data is from the 10 ANZA network stations, since the southern California network stations which are used do not have horizontal seismometers and have a more limited dynamic
range. The velocity inversion was constrained to have surface P wave velocity of 5.24 km/sec and surface S wave velocity of 2.84 km/sec based on the results of velocity measurements made in 300 meter deep
boreholes located at stations KNW and PFO [*gFletcher et al.*g, 1989]. The final one-dimensional model (Table 1) uses station corrections in the determination of velocities at a set
of depth nodes and assumes a linear gradient between these nodes. The calculated error in epicentral location has been reduced to 0.18 km and in depth to 0.43 km.

This improved velocity model can be used to generate ray paths which help to visualize the take-off angles from the earthquake sources and the expected angles of incidence as a function of source-receiver
distance. Figure 1 shows the ray paths through the velocity models in Table 1 for sources at 2.5, 5, 10, and 20 km depth. The same set of take-off angles
at each depth was used for both P and S waves. These plots illustrate, for includely all except the shallowest events, that the data collected will only sample seismic rays emanating from the upper half
of the focal sphere. The P and S waves each sample different parts of the crust, with the amount of path separation increasing as the hypocentral depth decreases. The difference of the P and S wave paths
is caused by the solutions to the velocity inversion not having a constant *v _{p} / v_{s}* ratio.

## 3. Seismicity Distribution

*gFletcher et al*g. [1987] identified four clusters of concentrated seismic activity in the study area shown in Figure 2. These clusters of events are: Hot Springs
(HS), Cahuilla (CA), Anza (AN), and Table Mountain (TM). There is also a region of diffuse seismicity between the surface traces of the Buck Ridge and Clark faults. Figure 2
shows the relocated epicenters for the whole database of events along with four cross sections. The depths of the events are measured from the top of Toro Peak (2657 m elev.), and seismic stations which
are near each cross section are displayed by triangles.

It is clear from the epicenter map and the cross sections in Figure 2 that the spatial distribution of events is not uniform. The seismicity is deepest near the surface
expression of the San Jacinto fault zone. Cross section A-A projects all events within 4 km into the plane of the San Jacinto fault zone. The AN and TM clusters are not part of this set of events. This
cross section shows that the base of the seismogenic zone rises from 24 km on the northwest to 18 km depth to the southeast. The section of the fault from the northwest end of the array to the trifurcation
of the fault zone is effectively aseismic from 0 to 12 km depth [*gSaunders and Kanamori*g, 1984; *Sanders*, 1986].

The deepest events in the study area are found in the HS cluster at depths from 15 to 24 km. These events, identified in cross sections A-A and B-B, are some of the deepest strike slip events in California. The HS cluster is located under the traces of the Hot Springs and the Clark faults. Except for a few isolated events, all the seismicity in the HS cluster occurs below 15 km depth. The B-B cross section does not define distinct fault planes, but displays that the events are progressively deeper to the north-east.

The shallowest events are found in cluster CA. Cross section C-C shows the depth of seismicity approaching 1 km depth below the surface (identified by the solid triangle). Most of the seismicity in this
cluster is located in a nearly vertical plane with a strike of about N10W. This cluster originally nucleated at a point as shown in Figure 4a of *gFletcher et al*g. [1987]. Since 1982 the seismicity
in this cluster has propagated in a northerly direction to define the plane of faulting.

The TM and AN clusters both appear to be fairly compact volumes of seismicity located at 10-14 km depth. The AN cluster (C-C) appears to be several kilometers southwest of the Clark fault [*gSanders*g,
1986]. The TM cluster (D-D), which has been active since at least 1968 [*gArabasz et al.*g, 1970], is also to the southwest of the fault zone. The diffuse region of seismicity under the trifurcation
area, while confined between 5 and 18 km depth, does not show any obvious spatial correlation between events.

Each of these clusters has distinct temporal signatures. Figure 3 plots the cumulative number of magnitude M_{L} ≥ 2.0 events which occurred in the HS, AN,
CA, and TM-Buck Ridge clusters. These events were located by the southern California seismic network between 1975 and 1989. The Hot Springs and Table Mt.-Buck Ridge regions, which both span the San Jacinto
fault zone, have a nearly constant rate of events per year. The Anza cluster had minimal activity until the June 15, 1982, M_{L} = 4.8 Anza earthquake. Since then there has been a slowly tapering
rate of aftershocks. The Cahuilla swarm had a low level of activity until several months following the M_{L} = 5.5 Whitewash earthquake on February 25, 1980, about 30 km to the east. For the next
year the Cahuilla cluster had a very high rate of activity which, by 1982, leveled off to a nearly constant rate. Remarkably, this rate is nearly identical to the Hot Springs cluster rate.

## 4. Earthquake Source Parameter Estimation Procedure

There are four types of earthquake source parameters which are routinely calculated for events recorded on the ANZA network. These parameters are the scalar seismic moment M_{0}, the source radius
r_{0}, the Brune stress drop δσ, and the a_{rms} stress drop δσ_{a rms}. The seismic moment M_{0} is a measure of the strength of an earthquake [*gAki
and Richards*g, 1980] and is defined

M_{0} = µ ü A

where µ is the shear modulus usually assumed to be 3 x 10^{11} dynes/cm^{2},

ü is the average slip over the fault plane,

and A is the area of the fault plane.

The source radius is an estimate of the dimension of the fault plane. The two estimates of stress drop are measures of the stress released by the slip during the earthquake.

These source parameters are calculated from several variables] determined from the far field seismic body waves. The seismic moment is linearly related to the low frequency displacement amplitude spectrum
asymtote Ω_{0} and is independent of the source model [*gAki and Richards*g, 1980]. The source radius is dependent on the corner frequency *f _{c}* of the displacement amplitude
spectrum. Based on the model of

*gBrune*g [1970, 1971], the stress drop δσ is a function of

*M*

_{0}and

*f*. The following equations from

_{c}*Fletcher et alg. [1987] are used for the calculation of the source parameters for the body wave arrivals at station*

*i*

M_{0 i}^{p, s} = (4ϖρ *v _{p, s}*

^{3}

*R*ω

_{i}_{0 i}

*) /*

^{p,s}*F*(θ, φ)

_{i}*r _{i}^{p, s}* = 2.34

*v*/ 2ϖ

_{p,s}*f*

_{ci}where ρ is the density of the rock,

*v _{p}* , Ω

_{0}

^{p}and

*v*, Ω

_{s}_{0}

^{s}are the velocity and low frequency displacement spectral level for P waves and S waves, respectively,

*R*is the hypocentral distance, and

_{i}*F*(θ, φ) is a radiation pattern correction term.

_{i}An alternative method to estimate stress drop was introduced by *gHanks and McGuire*g [1981] based on the root mean squared acceleration of the body wave phase

*a _{rms i}* = [ 1 / T

_{d}∫

_{0}

^{Td}a

_{i}

^{2}(

*t*)

*dt*]

^{1/2}

where *a* _{i }(*t* ) is the acceleration time series and *T _{d}* is the time duration of the shear wave.

δ σ_{ai} = 2.23ρ *R* [ -4/5 *f _{c}* + 2 η/2 η -1

*f*]

_{max}^{1/2}

*a*

_{ rms i}

*f*

_{c}^{1/2}, and

*f*<

_{c}*f*

_{max}where *f _{max}* is a signal limiting corner frequency in the acceleration spectrum probably due to a site effect [

*gHanks*g, 1982], and η is the order of the roll-off of the acceleration spectrum above

*f*. The formulae for the estimates of the earthquake source parameters are averages from the individual estimates at N stations

_{max}/ log *M*_{0} = ( ∑_{i=1}^{N} *w _{i}* log

*M*

_{0i}) / (∑

_{i=1}

^{N}w

_{i})

[bar] r_{0} = 1/*N* ∑_{i=1}^{N} *r* _{0 i}

[bar] δσ = 7/16 (M_{0}) / (*r* _{0}^{3} )

[bar] δ σ_{a} = 1/*N* ∑_{i=1}^{N} δ σ_{ai}

where *w _{i}* is a weight assigned for each pick by the analyst [

*gHaar et al.*g, 1984].

## 5. Earthquake Source Parameters

The database of events inside the study area in Figure 2 has a moment range of over four decades from 1.3 x 10^{10} N-m to 4.4 x 10^{14} N-m. For each
event a set of source parameters is calculated for both the P and S waves. The determinations of the P and S waves for the seismic moments vary by a factor of 3 from each other over four decades of moment
(Figure 4). A similar comparison of the source radius calculated from the P and S waves show a factor of 2 variation over a one decade range in Figure
4.

*gFletcher et al*g. [1987] found that the maximum stress drop of events appeared to be depth dependent. Both the Brune and the *a*_{rms} stress drops are shown as a function of depth
in Figure 5. The maximum stress drop for both estimates tend to increase with depth until 4 km is reached. For the next 16 km the maximum stress drop appears to remain constant
with the exception of several outliers below 11 km depth. Stress drops have a wide range in value for the whole depth distribution of events. Even for the deepest events there are stress drops of less than
l bar observed. The values for the *a*_{rms} stress drop are consistently higher than for the Brune stress drop. When the two estimators are compared in Figure
6 the *a*_{rms} values are, on average, a factor of 2 to 3 times larger than the Brune stress drop. The *a*_{rms} stress drop calculation assumes that the earthquake source
obeys the *w *^{-2} model of *gBrune*g [1970, 1971]. One explanation for the positive bias of the *a*_{rms} stress drop is that the seismic source displacement spectrum
has a roll-off which is less than -2, another is that the two sample windows in the two determinations are not consistent.

When the moments are plotted against the source radius (Figure 7) an interesting feature presents itself. For a moment range of 4 decades the earthquakes have a uniform
band of source radii which is bounded between 50 and 300 meters. This implies that earthquakes with magnitudes between M_{L} = 1 and M_{L} = 4 have essentially the same mean rupture area
and are differentiated primarily by the amount of slip [*gTucker and Brune*g, 1973; *Rautian et al.*, 1978; *Fletcher*, 1980]. This result must be examined carefully to ascertain whether this
is a real phenomena or is an artifact of our data collection or data processing methods (i.e., whether the observed corner frequency is a true source effect, or rather an effect of attenuation and site
response).

The source parameters which are being studied are based fundamentally on three different observables. The seismic moment is linearly related to the low frequency level of the displacement amplitude spectrum
Ω_{0}. The source radius is dependent on the corner frequency *f _{c}* of the displacement amplitude spectrum. The Brune stress drop is a function of both variables. The

*a*

_{rms}stress drop is a function of the integral of the acceleration time series for the body wave of interest and is dependent on the choice of the integration limits (0 -

*T*) by the interpreter.

_{d}For many events there are between 6 and 10 individual values to determine the source parameters. This falls deep into the realm of small sample statistics, but there is enough information to get an understanding
of the stability of the three independent observables of moment, source radius, and *a*_{rms} stress drop. Both the log ( *M*_{0} ) and the log ( *f _{c}* ) were
considered to be normally distributed by

*gFletcher et al*g. [1987]. The derivation for calculating the standard deviation of a log-normally distributed variable is given in the Appendix. Table 2 gives the mean value of the standard deviation and the coefficient of variation (Cv) which equals the standard deviation/mean for the aforementioned source parameters. For all of these variables Cv is large, which does not give a high degree of confidence to the averaged source parameter values. For the estimates of stress drop the values of Cv indicate that /δ σ

_{a rms}is the most stable.

Parameter | Average Coefficient of Variation (SD/MEAN) |
---|---|

f_{cp} |
0.35 |

f_{cs} |
0.44 |

M_{0p} |
0.85 |

M_{0s} |
0.79 |

δ σ_{a rms} |
0.94 |

δ σ_{P} |
1.38 |

δ σ_{S} |
1.45 |

## 6. Borehole Data

*One possible source of contamination for the estimation of source parameters is the effect of the local site structure. The velocity gradients found in the near surface weathering layers can amplify
or attenuate different sections of the seismic body wave spectrum. gFletcher et alg. [1989] describes the recent installation of two sets of borehole seismometers at 150 and 300 m depth along with
a surface sensor near the ANZA station KNW. This station was chosen because it is considered to be a "hard rock" station (Chapter II) with a high f_{max} value [gFletcher et
al.g, 1987] and is considered to be excellent for measuring high frequency seismic waves. The data were recorded on GEOS recorders [gBorcherdt et al.*

*, 1985] which uses a 16 bit analog-to-digital convertor and samples at 400 samples/sec.*

Twelve events were analyzed which were recorded on all three sets of sensors during a 20-day span in March 1988. These events have magnitudes from *M _{L}* = 1.4 to

*M*= 2.6. Most of these events are located in the Hot Springs cluster. The ANZA network was operating at a low gain so only one of the events recorded on the borehole was large enough to record on it. Displacement amplitude spectra were calculated for the body wave phases. Figure 8 gives an example of the displacement amplitude spectrum calculated for the P wave on the vertical component from a very small event. The sensors at 150 and 300 meters depth have a nearly flat displacement amplitude spectrum between 5 and 80 Hz. This figure reproduces the results of

_{L}*gHaar et al*g. [1989] who observed that that spectra recorded at this site on the surface sensors show strong amplifications in the frequency band 10-40 Hz. These amplifications can be explained by the mean surface velocity structure [

*gHaar et al*g., 1989]. They also noted that the spectra recorded on the surface are attenuated above 60 Hz with respect to the borehole spectra.

The low frequency displacement amplitude level, corner frequency, and Brune stress drop were calculated for the twelve events from the recordings of the three sets of sensors (Figure
9). Comparison of these three parameters for the P and S phases shows that all the moment estimates from the surface instruments are biased high by a factor of ~3 when compared to the estimates from
the borehole sensors after correcting for the free surface amplification of x 2. The values calculated from the two borehole recordings show an amplification of about 1.5 for Ω_{0} measured
on the 150-m sensors when compared to Ω_{0} determined from the 300-m sensor. When the corner frequencies for the P waves are examined, the estimates from the two borehole depths also approximate
a one-to-one relationship. The corner frequencies for the surface sensors show a preferred value for all the events for the P wave centered at 45 Hz.

The surface resonance raises the corner frequency for the P waves when *f _{c}* measured on the borehole is below 35 Hz and lowers them when

*f*on the boreholes are above 50 Hz. The data for the corner frequencies of the S waves shows a different trend than that for the P waves. There does not appear to be a significant relationship between deep and shallow corner frequency picks except for a tendency for the corner frequency pick from the deep sensor to be somewhat greater. The data for the S waves may be in error since the orientations of the borehole horizontal seismometers may not be accurately known. The corner frequencies used are the average between the two sensors. The comparison of the stress drop estimates from borehole and surface sensors shows that the surface measurements yield a slightly higher stress drop estimate for both the P and S waves.

_{c}## 7. Stress Drops

One of the topics of active controversy in seismology is whether the averaged stress drop is a constant value of 30-100 bars for all earthquakes, or whether the stress drop decreases with moment for
small events (*M*_{0} < 10^{13} N-m). The data recorded on the ANZA network (Figure 7) are not conclusive. The data, if taken at face value, suggest
that there are many small events with stress drops less than one bar. The standard deviation of each moment and source radius estimate is of the same order as the mean. However, with hundreds of events
the random errors will average out. This does not preclude systematic errors, such as corner frequencies being limited by site effects or attenuation. The attenuation at depth in the area of the ANZA network
appears to have values for Q of over 1000 for both P and S waves [*gHough and Anderson*g, 1988]. *Haar et al*. [1989] support this conclusion by observing that there is no evidence for *f*_{max}
measured on the borehole sensors at KNW, and concluding Q ≥ 4000. But there is evidence for site effects in the corner frequency picks. A plot of P and S wave corner frequency picks by station (Figure
10) shows that there are upper bounds for *f _{c}* which are different at every station. The propagation of the rupture front toward (or away from) an observer can cause an apparent corner
frequency shift similar to a Doppler effect. But the distribution of seismicity would suggest that source effects like this will average out with the observation of many events. The observation of a station
dependent maximum

*f*

_{max}suggests that this is a local site effect [

*gHanks*g, 1982]. If we consider only the stations LVA, SND, and TRO, none of them has had a source radius estimate of smaller than 70 meters for S waves and 60 meters for P waves during the entire operation of the array.

On the other hand, KNW, PFO and RDM have had source radius estimates which are over a factor of 2 smaller for both P and S phases. All three low corner frequency sites are considered to have low *f*_{max}
values.

Figure 11 shows the moment vs source radius data for the Hot Springs cluster from data recorded on the ANZA network. This cluster is closest to the KNW borehole and surface sites with most of its events located at a hypocentral depth of 15 to 24 km. Superimposed on this plot are the S wave moment and source radius estimates for twelve events which were recorded at the borehole site. The estimates from this site are presented in the form of a vector where the tail and the head represent the determinations from the surface and 300 meter depth sensors respectively. The corrections suggested by values calculated from the downhole data decrease the estimate of moment and usually decrease the source radius value as was also shown in Figure 9. These changes have opposing effects on the stress drop determination, however the reduction in moment has a greater influence for several of these events.

Another interesting feature presented in Figure 11 are the actual values of the estimates of moment and source radii. Each of the observed earthquakes are quite small,
yet the individual estimates from the borehole and the surface data yield values of stress drop which are significantly larger than the averaged values from the ANZA data. The estimate of source radius
from the network data are averaged over six or more independent stations and could be biased upward by the limits of the corner frequencies found at certain stations (Figure
10). *gFletcher et al*g. [1987] observed that if only stations with *f*_{max} > 30 Hz are used for determining source parameters then the smallest source radii measured from S waves
were between 30 and 40 meters for events with moments smaller than 2 x 10^{11} N-m. However, the data from the borehole site gives higher stress drops than this *f*_{max} selected
ANZA data of *gFletcher et al*g. [1987].

The increase in the stress drop estimates caused by the elimination of the low *f*_{max} sites from the calculation suggests that a closer examination of the individual station data is
warranted. Figure 12 plots the Hot Springs cluster data along with the individual station estimates from the KNW (ANZA) station. KNW is the closest station to the Hot Springs
cluster and has the advantage of being a high f_{max} station as well. The stress drops from KNW show a positive offset to the network averaged data. This effect is controlled by the smaller source
radius estimates.

The shift in source radius can be explained by two processes. The first possibility is the propagation properties of the transmission path such as intrinsic attenuation, scattering, or site effects like
different *f*_{max} values. If this is the case, then perhaps only the closest stations should be used for source parameter estimation. The other alternative is that we are observing a true
source effect which increases the corner frequency for stations directly over the hypocenter. Since this is not explained by current theoretical models, further detailed studies would be suggested.

## 8. Conclusions

From the set of events which have been recorded on the ANZA seismic network, seismic source parameters have been determined for 798 earthquakes located in the vicinity of the array between October 1982 and July 1988. These events have been used to determine the following:

- A joint hypocenter-velocity structure inversion using data from the ANZA and the southern California seismic networks has reduced the epicentral errors from 1.5 to 0.18 km and the depth errors from 3.0 to 0.43 km.
- The seismicity in the vicinity of the array forms several regions of diffuse events. Many events do not appear to be directly related to fault traces.
- Some of the deepest strike slip events in California occur in the Hot Springs cluster near the northwest edge of the array over 20 km below the surface.
- Source parameters estimated from P and S waves encompass a range of scalar seismic moment from 10
^{10}to 10^{15}N-m and a distribution of estimated stress drops from 0.1 to 200 bars. The source parameters have standard deviations which range from 40% to 140% of the mean value. - Measurements made from 300 meter deep borehole sensors at KNW yield moment estimates which are a factor of ~3 lower than surface measurements. This is caused by the amplification in the surface weathering layers of frequencies in the 10 to 40 Hz band.
- Source radius estimates appear to have a lower bound of approximately 25 meters for events with moments between 10
^{10}and 10^{15 }N-m. This could possibly be caused by a site*f*_{max}effect or be a true limit on the dimension of the earthquake source.

The question of whether the low stress drops for the low moment earthquakes shown in Figure 9 are a real earthquake source effect or are propagation and/or site effects is still unresolved. The limiting factor in the stress drop determination using data recorded on the surface is the source radius. Out of over 3000 source parameter picks from 500 earthquakes there are only thirteen which have values between 25 and 40 meters. The source radii calculated from the surface sensor data at the borehole site are between 36 and 60 meters, similar to the ANZA network data. The smallest source radii calculated from downhole data is 24 meters which is not significantly less than the smallest values found using the ANZA stations. This evidence suggests that there are events occurring with small moments and which have stress drops on the order of one bar. What is needed to disprove this assertion is data which shows that either the 300 meter deep sensors are still too close to the free surface and/or the surface weathering layer or that the high frequency part of the seismic signals is attenuated.

## Appendix

Define *l* such that to determine the variance of a log normal distribution

M over {*gE*g^"{" ^ M ^ "}"} ~=~ e^{l}

Assume $M$ is log-normal so that the probability distribution of *l* is

p_{e} ( l ) ~=~ 1 over {sqrt {2πσ_{e}^{2}} } ~exp~ left [ -~ {(l ^-^ *gE*g^"{" ^l^ "}" )^{2}} over {2 σ_{e}^{2}}
right ]

where $*gE*g^"{" ^l^ "}" ~=~ l hibar ~==~$ mean value of *l*.

The variance of $l$ is σ_{l}^{2} ~ mark =~ *gE*g^"{" (l ^-^ *E*^"{" ^l^ "}" )^{2} "}" lineup =~ *E*^left "{"
left [ log ~ left ( M over {*gE*g^"{" ^ M^ "}" } right ) ~-~ *E*^left { log ~ left ( M over {*E*^"{" ^M^ "}"} right ) right } right ]^{2} ^ right
"}"

Therefore, the expectation of (A1) gives *gE*g^left "{" M over {*E*^"{" ^M^ "}"} right "}" ~=~ *E*^"{" e^{l} "}" ~=~ 1

The average value of the function is found by integrating that function over the probability distribution.

*gE*g^"{" e^{l} "}" ~ mark =~ int from {-^inf} to inf ~ e^{l} p_{l} ( l ) dl lineup =~ 1 over {sqrt {2πσ_{l}^{2}} } ~ int ~ exp
left [ l ~-~ {l^{2} ^-^ 2l *gE*g^"{" l "}" ^+^ *E*^"{" l "}"^{2}} over {2 σ_{l}^{2}} right ] ~ dl

Completing the square in the exponent in the integrand

l ~-~ {l^{2} - 2l *gE*g^"{" l "}" + *E*^"{" l "}"^{2}} over {2 σ sub l^{2}} ~ mark =~ {-^l^{2} + 2l ( *E*^"{"
l "}" + σ_{l}^{2} ) - ( *gE*g^"{" l "}" + σ_{l}^{ 2} )^{2}} over {2 σ_{l}^{2}} ~+~ {( *E*^"{"
l "}" + σ_{l}^{2} ) ^{2} - *gE*g^"{" l "}"^{2} } over {2 σ_{l}^{2}} lineup =~ {-^ [l ^-^ ( *E*^"{"
l "}" ^+^ σ_{l}^{2 }) ]^{2}} over {2 σ_{l}^{2}} ~+~ {( *gE*g^"{" l "}" ^+^ σ_{l}^{2 })^{2}
^-^ *gE*g^"{" l "}"^{2} } over {2 σ_{l}^{2}}

using (A6) in (A5) gives

*gE*g^"{" e^{l} "}" ~ mark =~ exp left [ ^ {( *E*^"{" ^l^ "}" ^+^ σ_{l}^{2} )^{2} ^-^ *E*^"{" ^l^
"}"^{2}} over {2 σ_{l}^{2}} ^ right ] lineup =~ exp ^[*gE*g^"{" ^l^ "}" ^+^ half ^ σ_{l}^{2} ]

since $*gE*g^"{" e ^{l} "}" ~=~ 1$ from (A3). Then

*gE*g^"{" l "}" ~=~ -^ half ^ σ sub l^{2}

From (A1) log ^ (M) ~-~ log ^ *gE*g^"{" M "}" ~ mark =~ l *E*^"{" log ^ M "}" ~-~ log ^ *E*^"{" M "}" ~ lineup =~ *E*^"{"
^l^ "}" lineup =~ -^ half ^ σ_{l}^{2}

From (A1) define the normalized variance

σ_{N}^{2} ~=~ {σ_{M}^{2}} over {*gE*g^"{" M "}"^{2}} ~=~ {*E*^"{" [M ^-^ *E*^"{" M "}"
]^{2} "}"} over {*gE*g^"{" M "}"^{2}} ~=~ *E*^left "{" ^ left [ ^ M over {*E*^"{" M "}"} ~-~ 1 ^ right ]^{ 2}
right "}" ~=~ *gE*g^left "{" ^ left ( ^ M over {*E*^"{" M "}"} right ) ^{2} -^1 right "}"

Using (A2) and integrating using the same procedure for (A5)

*gE*g"{" e^{2}l "}" ~=~ int from {- inf} to inf ~ e^{2}l ~ p_{l} ( l ) ~dl ~=~ e^{{σl} }

therefore

σ_{N}^{2} ~=~ e^{{σl2} ~-~1}

or

σ_{l}^{2} ~=~ log ^ (1 ^+^ σ_{N}^{2} )

From (A12) and (A16)

*gE*g"{" log ^ M "}" ~-~ log ^ *E*"{" M "}" ~=~ -^ half ^ log^ ( 1 ^+^ σ_{N}^{2} )

log ^ (1 ^+^ σ_{N }^{2} ) ~=~ 2(- *gE*g"{" log ^ M "}" ~+~ log ^ *E*"{" M "}" )

to reach

σ_{N}^{2} ~=~ {σ_{M}^{2}} over {*gE*g"{" ^M^ "}"^{2}} ~=~ exp [ ^-2 *E*"{" log^ M "}" ^+^ 2^log^ *E*"{"
M "}" ] ~-~ 1

σ_{M }^{2} ~=~ *gE*g"{" ^M^ "}"^{2} ~ [ exp ( ^2(- *E*"{" log^ M "}" ^+^ log ^ *E*"{" M "}" ))
~-~ 1]

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