### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Explosion and Damage Source Modeling
- 3 Results
- 4 Discussion and Conclusions
- Acknowledgments
- References

[1] Effects of rock damage on teleseismic *m*_{b} are investigated with *P* wave synthetic seismograms using a moment dipole *M*_{zz} as the equivalent elastic model for damage around buried explosions. Two manifestations of late-time damage, cavity rebound and bulking from block rotations, are represented by model decompositions into compensated linear vector dipole and monopole sources, respectively. For high-velocity media, *P* waves from damage destructively interfere with those from the explosion. This interference reduces the rate at which *m*_{b} yield scales for a pure monopole source and provides a physical basis for observed scaling in hard rock, *m*_{b} ~ 0.75 log[yield]. For over-buried explosions, such as the North Korean tests, *P* waves from damage are weaker, and higher scaling rates are expected than explosions conducted under standard containment conditions. Our results highlight a cautionary note of transporting the same *m*_{b}-log[yield] relation between test sites to estimate yield when source phenomenology is likely to be very different.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Explosion and Damage Source Modeling
- 3 Results
- 4 Discussion and Conclusions
- Acknowledgments
- References

[2] A recent study [*Patton*, 2012] shows that “late-time” damage can affect the apparent yield scaling of *M*_{s} for underground nuclear explosions. This damage refers to various modes of material failure occurring after shock waves have encountered the free surface, and the stress regime of the source medium becomes tensile. In contrast, “prompt” damage occurs under compression as the initial shock wave travels outward from the source [e.g., *Ashby and Sammis*, 1990]. The late-time phenomenology combined with the effects of shock-wave rebound [*Jones et al*., 1993] can be a significance source of seismic waves.

[3] But what about *P* waves radiated by late-time damage and their effect on *m*_{b}? This paper investigates the possibility that *P* wave radiation from damage affects the apparent yield scaling of teleseismic *m*_{b}. The damage model in this paper captures medium damage during the time from the *pP* reflection on in the short-period passband primarily affecting the second upswing (the “*c*” amplitude) of the *P* wave. As *m*_{b} usually involves a peak-to-peak measure, the amplitudes tend to get reduced by damage due to weakening of the second upswing, and this effect increases with depth of burial (or yield under standard containment practices) reducing the *m*_{b} scaling coefficient.

[5] These findings pose a conundrum for observations of hard rock explosions detonated at the Balapan test site in Semipalatinsk and elsewhere. A scaling rate of 0.75 for *m*_{b} is firmly established in the literature for such explosions [e.g., *Ringdal et al*., 1992; *Murphy*, 1996, to name a few prominent studies]. How can this well-accepted scaling rate be reconciled with the fact that *pP* interference significantly increases the *m*_{b} versus yield scaling to values 1.0 or more? In this paper, we will show that it is possible for waves radiated from the damage source to destructively interfere in the *P* wave window and reverse the effect on *m*_{b} scaling due to *pP* for a pure explosion. This result has important implications for the transportability of *m*_{b}-log[*W*] relationships between test sites. Specifically, it questions the validity of applying a relationship for Balapan nuclear explosions to North Korean explosions for yield estimation purposes when the containment conditions (and thus damage sources) may have been quite different for tests at the two sites.

### 2 Explosion and Damage Source Modeling

- Top of page
- Abstract
- 1 Introduction
- 2 Explosion and Damage Source Modeling
- 3 Results
- 4 Discussion and Conclusions
- Acknowledgments
- References

[6] A synthetic seismogram code was prepared to perform calculations following the technique of *Kennett* [1983]. The advantage of using this technique is that the up-going and down-going energies can be separated. This is important when the effect of rock damage is included. The basic equation to synthesize *P* wave motion *u*_{p} in a half space is given by

- (1)

[7] Reading from right to left, the factors in Kennett's notation are the down-going waves from a source *s*, the transmission matrix for the down-going waves to the buried receiver *r*, plus (+) the up-going waves from a source, the up-going transmission from a source to the free surface, the free surface reflection coefficient, and finally the down-going transmission from the free surface to the buried receiver. The latter term gives the *pP* arrival and the former term the direct *P* arrival at receiver *r* located below source *s*. The reflection coefficient (*R* in equation (1)) is a simple way to represent reduction in *pP* amplitude due to topography and near-surface scattering.

[8] In this paper, the damage contribution will be represented with a vertical dipole force system [*Knopoff and Randall*, 1970; *Ben-Zion and Ampuero*, 2009; *Patton and Taylor*, 2011 (hereinafter referred to as PT11)]. The dipole can be decomposed into two components, one volumetric and one deviatoric in the form of a Compensated Linear Vector Dipole (CLVD).

- (2)

[9] The volumetric component is thought to be due to macro- or micro-scale bulking from block rotation and shear dilatancy [*Heuzé et al*., 1991]. The CLVD is related to shock-wave rebound around the waist of the explosion, which is redirected upward in response to a developing tensile regime from free-surface interactions [e.g., *Jones et al*., 1993; PT11]. Gravitational unloading of spallation can enhance the deformations from both components. For highly over-buried explosions, the effects of this damage scenario are thought to be reduced or absent [e.g., *Taylor*, 2009]. Assuming linear superposition, the damage in equation (2) gets added to the explosion monopole and can actually increase the net volumetric moment beyond the effects of the explosion creating a cavity and can affect yield estimation (PT11).

[10] The relative strength between the explosion monopole and damage source is governed by the *K* parameter defined in PT11:

- (3)

[11] For a spherical explosion represented by a monopole with unit moment, the weighting for the damage source written in equation (2) is given by *K* − 1 [*Patton*, 2012]. An impulse moment-rate time function is used for the explosion monopole. For the damage source, a simple moment-rate time function of the form

- (4)

is used where *τ* represents the characteristic time of the damage process. The damage source is triggered by the passage of the *pP* from the monopole computed using the elastic travel time.

[12] A schematic of the model is shown in Figure 1. Free parameters are the depth of burial *h*_{x}, the centroid depth of damage *h*_{d}, the characteristic time *τ*, the surface reflection coefficient *R*, the relative source strength *K*, the *P* wave speed *V*_{p} of the Poisson medium, and the rock density. The synthetics are passed through a World Wide Standard Seismographic Network (WWSSN) Short Period (SP) instrument. For all of the cases described below, *R* was set to 0.3 and *h*_{d}/*h*_{x} to 0.5. The dependence of our results on these parameters will be investigated in the future.

[13] One reason for choosing the WWSSN SP passband is that our continuing work of explosion source physics will require close examination of historic nuclear explosions detonated under different emplacement conditions. Many of these explosions are old enough that the majority of recordings were on the WWSSN. That being said, future work will include a variety of *m*_{b} measurement procedures in order to examine variations between different magnitude scales.

[14] Over time, procedures for measuring *m*_{b} from explosions have adapted to accommodate waveform complexities. Following Blacknest Seismological Center procedures [*Stewart*, 1988], automated measurements were made to compute *a*, *b*, and *c* amplitudes and times as well as the *T*_{ox} zero-crossing time. Figure 2 shows synthetic teleseismic *P* waves and measurements used to compute *m*_{b} = log(*A*/*T*) for two cases, *K* = 1 (pure monopole) and 3 with *V*_{p} = 5100 m/s and *h*_{x} = 550 m. The amplitude (*A*) is *A*_{bc} where the *b* and *c* amplitudes and times are given by green and blue dots, respectively. For a simple waveform shown in Figure 2a, the period *T*_{bc} is twice the time between *b* and *c* amplitudes. When there is an inflection in the *b-c* swing (Figure 2b), *T*_{ox} is used for period. The presence of an inflection is determined from the second derivative of the waveform.

### 3 Results

- Top of page
- Abstract
- 1 Introduction
- 2 Explosion and Damage Source Modeling
- 3 Results
- 4 Discussion and Conclusions
- Acknowledgments
- References

[15] A large number of runs were made using ranges of *K* from 1 to 3, *V*_{p} from 4000 to 6000 m/s, and *h*_{x} from 250 to 600 m. For a standard containment depth of 120 scaled meters, the burial depths correspond to a yield range of ~8 to 125 kT. No explosion source time function was used because our calculations show that the frequency dependence will be small through the WWSSN SP passband for this yield range.

[16] Figure 3a is a three-dimensional (3-D) plot of log(*A*/*T*) for the explosion monopole as a function of *V*_{p} and *h*_{x} using *T*_{bc} because all waveforms were simple without inflection points. In general, the effect of *pP* interference increases the dependence of log(*A*/*T*) with *h*_{x}. For each *V*_{p}, linear regressions were performed on log(*A*/*T*) measurements as a function of log yield. Yield was computed from *h*_{x} using a containment practice 120 m/kT^{1/3}. The estimated slopes plotted against *V*_{p} in Figure 3b represent perturbations due to *pP* interference from spectral predictions of *P* wave amplitude scaling. As discussed in the section 1, the yield-scaling exponent is ~0.75 for Mueller-Murphy spectra. Clearly, the effect of *pP* interference is to increase the yield scaling, a result consistent with *Stevens and Day* [1985, Figure 7]. Above velocities of 4600 m/s, scaling exponents exceed 1.0, and for a *V*_{p} of 5100 m/s, the speed adopted by *Koper et al*. [2008] for the North Korean test site, the yield slope exceeds 1.1.

[17] Figure 4a shows the 3-D plot for *K* = 2.2 and *τ* = 0.1 s. The regression results for *K* ranging from 2.1 to 2.5 are shown in Figure 4b. This range was chosen based on inferences of *K* values from analysis of the effects on *M*_{s} for selected Balapan explosions [*Patton*, 2012]. The effect of damage for hard rock media (*V*_{p} > 4600 m/s) is to reduce the yield-scaling exponent from what it would be for a pure explosion with *pP* interference.

[18] This is further explored by examination of waveforms in Figure 5. The reason for the decrease in the *m*_{b} scaling can be seen by the sequence of plots for increasing *h*_{x} where the *P* wave from the monopole is shown with a red line, the *P* wave from damage with a blue line (*K* = 2.2) and the resultant wave form with a black line. The predominant effect is for *pP* from the damage source to destructively interfere with the *c* amplitude from the monopole. This interference can also affect the period measurement used for computing *m*_{b} further complicating the situation. As *h*_{x} increases, the effect of the damage source reducing the *c* amplitude of the resultant waveform gains efficiency due to better phase alignment of the *b* amplitude from damage and the *c* amplitude from the monopole.

### 4 Discussion and Conclusions

- Top of page
- Abstract
- 1 Introduction
- 2 Explosion and Damage Source Modeling
- 3 Results
- 4 Discussion and Conclusions
- Acknowledgments
- References

[19] In this paper, *K* is assumed to be constant for all burial depths *h*_{x}. Since *h*_{x} is a surrogate for yield, this assumption implies yield independence. In contrast, measurements of *K* for explosions conducted on the Nevada Test Site (NTS) show strong yield dependence (PT11). The interpretation is that the force of spall slap down steady increases with yield and eventually reaches a strength threshold of the source medium. Above this threshold, the medium fails in compression, crushing pre-existing pores in the volcanic rocks and eliminating voids created earlier in the damage process, thereby reducing the net static moment due to damage and thus *K*. In hard rock media, such as granite, pre-existing porosity is low, and the strength of the rock matrix is greater than it is for volcanics. Thus, *K* is expected to show less yield dependence for explosions in hard rock. Inferences about yield dependence of *K* for Balapan explosions [*Patton*, 2008] suggest it is weak.

[20] A *τ* of 0.1 s is the smallest value we tried, and 0.2 s gave only minimal perturbations on the scaling slope. While *τ* affects group/phase delay, the principal effect of longer source duration is to reduce the *P* wave amplitude. Of course this is all relative to the selected *K* values and the source spectrum of the monopole. For this preliminary look at the effect of damage, this spectrum was assumed to be white; on-going studies will introduce a realistic source function for the monopole. It is interesting to speculate that *τ* might depend on material properties: a longer *τ* in gas-filled, porous rocks like at NTS due to strong attenuation of the up-going shock waves than at granite sites like Balapan where the water table is close to the free surface. The functional form of the damage time function will also be investigated in future studies.

[21] *Murphy* [1996] summarizes *m*_{b}-log[*W*] scaling observations for some PNEs. This is of interest since the damage source should be relatively unimportant for PNEs that were significantly over-buried for their yield. Based on the results in our Figure 3, the scaling slope for *m*_{b} is expected to be greater than 1.0 for source media with *V*_{p} > 4800 m/s. The reader is referred to Figure 3 of *Murphy* [1996, p. 229], where regression results for eight over-buried PNEs conducted in limestone/dolomite gave an estimated scaling slope of 1.05 (uncertainties on this estimate were not reported). While supportive of results in our Figure 3, the interpretation of Murphy's observations remains ambiguous until the effects of *pP* on estimates of *m*_{b} for deeply buried explosions (both in terms of actual and scaled depths) are better understood. This is another area of future study.

[22] Concerning transportability of *m*_{b}-log[*W*] relations for yield estimation purposes, an understanding of coupling and upper mantle attenuation effects on the intercept was key to resolving controversies surrounding the verification of the Threshold Test Ban Treaty (TTBT) in the 1980s and 1990s. It should be noted that similar containment practices carried out by the United States and former Soviet Union testing programs likely mitigated the effects on yield scaling. As shown in this study and others, such practices impact the scaling of *m*_{b} (and *M*_{s}). In contrast to the TTBT with a set threshold of 150 kT, the transportability of both intercept and scaling parameters is critical for accurate yield estimation over a wide range of source sizes.

[23] Clearly, a physical basis for transporting both parameters must be understood for situations where different containment rules are followed. This has implications for the North Korean tests. *Patton and Taylor* [2008] show that the cause of poor *m*_{b}–M_{s} discrimination performance can be explained by a suppressed damage source, probably because these tests were over-buried and conducted in media that was strong and intact. As shown in Figure 3b, a wave speed of 5100 m/s adopted by *Kope*r *et al*. [2008] for the North Korean test site predicts a scaling slope of ~1.1 for a pure monopole. This result highlights a cautionary note of simply applying the hard rock *m*_{b}-log[*W*] relation based on Balapan experience to estimate the yields of these tests when the source phenomenology at the two sites is likely to be very different.

[24] In conclusion, synthetic seismogram calculations show that interference of the damage signal with that of the explosion monopole can affect *m*_{b} measurements and yield scaling through amplitude and period perturbations. This can have a significant impact upon the ability to estimate the yield of underground nuclear explosions. The models used in this paper are simplistic, and numerous enhancements can be made to better represent the true physics of the explosion source based on knowledge from close-in measurements and calculations. The two components comprising the *M*_{zz} representation of damage are quite distinct and probably occur at different depths with different characteristic times. The damage components are also distributed sources, but this may not be a factor for frequencies associated with *m*_{b} measurements. Rather, it may be possible to continue to use point source representations having different characteristic time functions.

[25] The purpose of this study has been to estimate the effect of the depth phases *pP* and *sP* on *m*_{b} for the combined explosion and damage source model. Is it possible that the destructive interference between *pP* from the monopole and the direct *P* wave from the damage source explains the so-called “*P* wave paradox” of *Douglas* [1991] and *Lay* [1991]? Can this interference effect also confound *m*_{b}–M_{s} discrimination? These questions and the exciting possibility that *P* waveforms may provide diagnostics about rock damage will require further research.