Corresponding author: S. R. Taylor, Rocky Mountain Geophysics, 167 Piedra Loop, Los Alamos, NM 87544, USA. (email@example.com)
 Effects of rock damage on teleseismic mb are investigated with P wave synthetic seismograms using a moment dipole Mzz 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 mb yield scales for a pure monopole source and provides a physical basis for observed scaling in hard rock, mb ~ 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 mb-log[yield] relation between test sites to estimate yield when source phenomenology is likely to be very different.
 A recent study [Patton, 2012] shows that “late-time” damage can affect the apparent yield scaling of Ms 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.
 But what about P waves radiated by late-time damage and their effect on mb? This paper investigates the possibility that P wave radiation from damage affects the apparent yield scaling of teleseismic mb. 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 mb 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 mb scaling coefficient.
Bache  shows that the far-field spectrum of P waves scales as (ρVp)0.5 (ω) where ρ and Vp are density and P wave speed, respectively, and (ω) is the spectrum of the reduced velocity potential. For the Mueller-Murphy model [Mueller and Murphy, 1971], it can be shown that (ω) scales with yield W as ~W0.75 for ~8 < W < 125 kT and a uniform granite source medium in the 1–2 Hz frequency band used to measure mb. Stevens and Day  took the next step by studying the effect of pP on time-domain amplitudes. They show that pP interference reduces P wave amplitudes used to measure mb of small explosions and increases amplitudes for large explosions buried at standard containment depths in granite. As such, pP interference increases the yield scaling of mb over what it is when pP is completely suppressed, e.g., mb ~ 0.75 log[W], the scaling of the source spectrum. In this study, we confirm this finding of Stevens and Day and show that pP interference can increase the apparent yield scaling of mb to 1.0 or greater.
 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 mb 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 mb 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 mb scaling due to pP for a pure explosion. This result has important implications for the transportability of mb-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
 A synthetic seismogram code was prepared to perform calculations following the technique of Kennett . 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 up in a half space is given by
 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.
 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).
 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).
 The relative strength between the explosion monopole and damage source is governed by the K parameter defined in PT11:
 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
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.
 A schematic of the model is shown in Figure 1. Free parameters are the depth of burial hx, the centroid depth of damage hd, the characteristic time τ, the surface reflection coefficient R, the relative source strength K, the P wave speed Vp 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 hd/hx to 0.5. The dependence of our results on these parameters will be investigated in the future.
 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 mb measurement procedures in order to examine variations between different magnitude scales.
 Over time, procedures for measuring mb 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 Tox zero-crossing time. Figure 2 shows synthetic teleseismic P waves and measurements used to compute mb = log(A/T) for two cases, K = 1 (pure monopole) and 3 with Vp = 5100 m/s and hx = 550 m. The amplitude (A) is Abc 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 Tbc is twice the time between b and c amplitudes. When there is an inflection in the b-c swing (Figure 2b), Tox is used for period. The presence of an inflection is determined from the second derivative of the waveform.
 A large number of runs were made using ranges of K from 1 to 3, Vp from 4000 to 6000 m/s, and hx 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.
 Figure 3a is a three-dimensional (3-D) plot of log(A/T) for the explosion monopole as a function of Vp and hx using Tbc because all waveforms were simple without inflection points. In general, the effect of pP interference increases the dependence of log(A/T) with hx. For each Vp, linear regressions were performed on log(A/T) measurements as a function of log yield. Yield was computed from hx using a containment practice 120 m/kT1/3. The estimated slopes plotted against Vp 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 Vp of 5100 m/s, the speed adopted by Koper et al.  for the North Korean test site, the yield slope exceeds 1.1.
 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 Ms for selected Balapan explosions [Patton, 2012]. The effect of damage for hard rock media (Vp > 4600 m/s) is to reduce the yield-scaling exponent from what it would be for a pure explosion with pP interference.
 This is further explored by examination of waveforms in Figure 5. The reason for the decrease in the mb scaling can be seen by the sequence of plots for increasing hx 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 mb further complicating the situation. As hx 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
 In this paper, K is assumed to be constant for all burial depths hx. Since hx 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.
 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.
Murphy  summarizes mb-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 mb is expected to be greater than 1.0 for source media with Vp > 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 mb for deeply buried explosions (both in terms of actual and scaled depths) are better understood. This is another area of future study.
 Concerning transportability of mb-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 mb (and Ms). 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.
 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  show that the cause of poor mb–Ms 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 Koper et al.  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 mb-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.
 In conclusion, synthetic seismogram calculations show that interference of the damage signal with that of the explosion monopole can affect mb 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 Mzz 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 mb measurements. Rather, it may be possible to continue to use point source representations having different characteristic time functions.
 The purpose of this study has been to estimate the effect of the depth phases pP and sP on mb 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  and Lay ? Can this interference effect also confound mb–Ms discrimination? These questions and the exciting possibility that P waveforms may provide diagnostics about rock damage will require further research.
 H.J.P. performed this work under the auspices of the Department of Energy for the Los Alamos National Laboratory under the contract DE-AC52-06NA25396. SRT performed this work under Subcontract Number 72276-001-09 to Los Alamos National Laboratory. We also would like to dedicate this work to Peter Marshall of the Blacknest Seismological Center (aka Lord Marshall of Newbury) who recently passed away.