A Gaussian explosion seismic energy source David F. Aldridge*, Thomas M. Smith, and S. Scott Collis, Sandia National Laboratories Summary An analytical expression for the pressure wavefield radiated from an explosion seismic source with an amplitude distribution in the form of a 3D Gaussian function is developed. This expression provides a useful reference solution for validating various numerical seismic wave propagation algorithms. Introduction A point seismic energy source, with spatial support given by a three-dimensional (3D) Dirac delta function, is commonly utilized in seismic wave propagation theory and numerical modeling. However, there is increasing interest in spatially-extended sources, perhaps because they constitute more realistic representations of physical seismic sources. Aldridge (2002) summarizes an extensive body of literature dating back to Jeffreys (1931) and Sharpe (1942) treating the classic “pressurized spherical cavity” source. More recently, Blair (2007, 2010) has re-examined and improved Heelan’s (1953) elastic radiation solution for a finite-length explosive column. A common aspect of these sources is that they are mathematically characterized as time-varying boundary conditions applied to the surface of a medium. In this investigation, we adopt the alternative point of view that a spatially-extended source may be considered a body source of seismic waves, and is thus represented by inhomogeneous terms in the governing system of partial differential equations. In effect, the source energy is instantaneously “deposited” within a portion of the medium supporting wave propagation. This may be a reasonable approximation in certain seismic source scenarios, as with large chemical or nuclear explosions. We develop a closed-form mathematical expression for the pressure wavefield generated by a spatial source with an amplitude distribution in the form of a 3D symmetric Gaussian function. Hence, this source has an infinite region of spatial support, although the magnitude diminishes to very small values more than about three standard deviations from the center point. The derivational methodology utilizes a representation theorem and a Green function for an isotropic elastic wholespace. The analytical solution for a Gaussian-distributed source forms an important reference solution for validating seismic wave propagation algorithms. In particular, in the discontinuous Galerkin (DG) numerical formalism, a 3D polynomial basis is used to approximate body source terms within a discrete computational element. The continuously-

differentiable Gaussian function is far more amenable to such polynomial approximation than a spatial Dirac delta function. Attempts to represent a Dirac pulse on a polynomial basis often lead to undesirable “ringy” oscillations. Governing Partial Differential Equations Within an isotropic elastic medium, the particle velocity vector vi(x,t) and the stress tensor �ij(x,t) satisfy a set of nine, coupled, inhomogeneous, first-order partial differential equations called the velocity-stress system. In rectangular coordinates xi (i = 1,2,3) these equations are

��

���

�∂

∂+=

∂

∂−

∂

∂

j

aij

ij

iji

x

mf

xt

v

ρ

σ

ρ11

, (1a)

t

m

x

v

x

v

x

v

t

sij

i

j

j

iij

k

kij

∂

∂=

∂

∂+

∂

∂−

∂

∂−

∂

∂���

����

�µδλ

σ. (1b)

δij is the Kronecker delta symbol, and summation over repeated subscripts is implied. The elastic medium is characterized by mass density ρ(x) and Lamé parameters λ(x) = ρ(x)[α(x)2−2β(x)2] and µ(x) = ρ(x)β(x)2, where α(x) and β(x) are the P and S wavespeeds. Inhomogeneous terms in (1a,b) represent body sources of seismic waves: fi(x,t) and mij(x,t) are components of the force density vector and moment density tensor, respectively. Note that the moment density tensor is split into symmetric and anti-symmetric parts, indicated by superscripts ‘s’ and ‘a’. The 3×3 moment density tensor forms a compact mathematical representation of a large variety of seismic energy sources, such as dipoles, couples, shear and tensile dislocations, explosions, torques, etc. The moment density tensor for a point source located at position xs is proportional to a 3D Dirac delta function, and is given by

)()(),( sijij twMdtm xxx −−= δ , (2)

where M is a moment magnitude scalar (SI unit: N-m = J), dij is a dimensionless orientation tensor, and w(t) is a dimensionless source activation waveform. It is common to normalize the orientation tensor and the wavelet via:

1=ijij dd , 1)(max =twt

.

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Gaussian ball explosion source

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Let an elastic wavefield be generated by the spatially-extended moment density tensor

���

����

����

����

� −−−=

2

3 exp1

)(3

),(hh

twMtm sijij

xxx π

δ. (3)

The isotropic orientation tensor dij = �ij /�3 implies that this body source is an “explosion” (i.e., represented by three mutually orthogonal force dipoles of equal magnitude). However, the spatial support for the source extends over the whole of the volume occupied by the elastic medium, rather than concentrated at a point. In the limit as the Gaussian width parameter h vanishes, equation (3) approaches the form (2). Time-Domain Pressure Derivation For mathematical tractability, specialize to a homogeneous and isotropic elastic wholespace. Wavefield representation theory (e.g., Aki and Richards, 1980) yields an expression for the pressure at position x and time t due to the Gaussian-distributed moment density tensor (3) centered at the origin xs = 0:

)( );,(ˆexp1

)(),(

2

3 xxxx

x ′′′

−∗= ���

����

�� dVtp

hhtMwtp

V

π .

(4) The asterisk denotes temporal convolution, and the volume integral is evaluated over the infinite space occupied by the elastic body. The impulse response pressure (SI unit: 1/(m3-s) generated by a unit magnitude point explosion placed at x and activated by a Dirac delta function is

( )���

����

� −′−′′

−′

−=′

αδ

πα

γ xx

xxxx ttp

1

4

341);,(ˆ

2

2

, (5)

where the P-to-S wavespeed ratio is � = �/�, and a prime denotes differentiation with respect to an argument. Combining equations (4) and (5) and evaluating the volume integral yield an expression for the pressure radiated from the Gaussian ball explosion source. The mathematical details of the approach are somewhat involved, and thus are briefly summarized as follows: 1) Fourier transform (4) and (5) from time t to angular frequency � and combine. 2) Parameterize the volume integral in terms of spherical polar coordinates (�, �, R) centered at receiver position x. 3) Evaluate the two integrals over the angular coordinates. 4) Inverse Fourier transform the resulting expression from frequency � back to time t.

5) Evaluate the remaining integral over spherical polar radius R. The result is a convolutional expression for time-domain pressure given by

),;,(),(),(pointGaussian

αhtrgtrptrp ∗= , (6)

where a “Gaussian explosion filter” is defined by

�

��

�

���

����

���−=

2

exp),;,(h

t

hhtrg

απ

αα

��

���

�

��

���

��

�

���

���

���

�+

+−−

α

ααπ

rtH

h

rt

)2(exp

2

, (7)

and H(t) is the Heaviside unit step function. Radius r = ||x|| is the distance from the coordinate origin to the field point x where pressure is observed. Note that this filter is causal with respect to the negative time –r/�, as is intuitively expected. It is straightforward to demonstrate that g(r,t;h,�) is a distribution sequence for the temporal Dirac delta function:

)(),;,(lim0

thtrgh

δα =→

.

Thus, as the width of the Gaussian ball diminishes to zero, equation (6) predicts that the pressure approaches that radiated by a point source, as required. This point source pressure is (Aldridge, 2000)

( )���

��� −′′

−=

απαγ r

twr

Mtrp 2

2

point 4

3/41),( , (8)

which is just (5) multiplied by magnitude M and convolved with wavelet w(t). Frequency-Domain Explosion Filter An important byproduct of the derivation is an expression for the frequency spectrum of the Gaussian explosion filter, expressed in terms of dimensionless arguments as follows:

( ) ( )[ ] ×−= exp2

1,

2cc

rrc

G ωωωω

[ ] [ ]��

���

�

��

�

���

����

����

����

����

� ∗Φ−−−−Φ− )(122

exp)(1 zr

riz

cc ω

ω, (9)

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Gaussian ball explosion source

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where )/()/( cc irrz ωω+≡ and the asterisk denotes

complex conjugation. �(z) is the error function with complex argument. Finally, �c and rc are characteristic filter parameters (angular frequency and range) given by

hfc

c

2 π

απ

ω== ,

π

hrc = .

Numerical evaluation of the spectrum presents some problems. Algorithm WOFZ (Poppe and Wijers, 1990) works well for ranges r/rc < 5. For r >> rc, equation (9) approaches the simple range-independent Gaussian filter

( ) ][ 2exp)( ccG ωωωω −= . (10)

Thus, at far-field distances, a spatially-extended Gaussian explosion is equivalent to a low-pass frequency filter applied to the analogous point source traces. Figure 1 depicts four normalized amplitude spectra. Rapid convergence to form (10) is evident.

Figure 1. Normalized filter amplitude spectra for r = 1, 25, 50, and 100 m (top to bottom). Time-Domain Pressure Signal Example Consider a homogeneous and isotropic elastic wholespace with P-wave speed � = 2500 m/s, S-wave speed � = 1500 m/s, and mass density = 2000 kg/m3. Figure 2 displays point source pressure traces obtained by evaluating equation (8) for ranges r = 5 m to r = 250 m. The source magnitude is M = 1 N-m, and the source wavelet is a Gaussian pulse given by

][ 2)(exp)( tpftw π−= , (11)

where fp = 30 Hz is the peak frequency of the “equivalent Ricker wavelet” obtained by double-differentiation. Pressure pulses exhibit linear moveout across the recording array as well as 1/r amplitude loss from spherical

divergence. The shape of the pressure pulse, at all ranges, is a Ricker wavelet (second derivative of a Gaussian wavelet). Figure 3 illustrates a set of time-domain Gaussian explosion filter responses calculated over the same set of source-receiver ranges. The width parameter is intentionally set to be rather large at h = 50 m, implying the characteristic frequency of each filter is fc = 28.2 Hz. The thin red line t = –r/� represents the causal onset time of each filter. An abrupt onset is clearly visible at small ranges, whereas beyond about r~75 m each filter is indistinguishable from a Gaussian pulse with a smooth onset. The characteristic range in this case is rc = 28.2 m. Finally, figure 4 displays pressure traces generated by a Gaussian explosion source centered at the coordinate origin, and with width h = 50 m. Each trace is obtained as the discrete convolution sum of corresponding traces in figures 2 and 3. Clearly, these Gaussian explosion-calculated traces differ in both amplitude and waveform from the analogous point explosion traces of figure 2. Pulse amplitudes are generally lower. Far-field pulse shapes are indistinguishable from a Ricker wavelet. Basic signal theory predicts the peak frequency of this far-field Ricker is

222

111

cpfar fff+= ,

which evaluates to ffar = 20.6 Hz in this case. The lowpass filter effect of the Gaussian explosion is evident. Conclusions and Ongoing Work An analytical solution for the pressure wavefield generated by an explosion seismic energy source with a 3D amplitude distribution in the form of a symmetric Gaussian function has been developed. The most significant effect of this spatially-extended source appears as a low-pass frequency filter applied to the analogous point source responses. The derivational approach is currently being extended to the particle velocity vector. The solution provides a useful reference for validating numerical seismic wave propagation algorithms. Preliminary comparisons with pressure responses generated by finite-difference and discontinuous Galerkin algorithms (not shown here) appear promising. Acknowledgment Sandia National Laboratories is a multiprogram science and engineering facility operated by Sandia Corporation, a Lockheed-Martin company, for the US Department of Energy’s National Nuclear Security Administration, under contract DE-AC04-94AL850.

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Gaussian ball explosion source

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Figure 2. Pressure traces recorded at ranges r = 5 m to r = 250 m generated by a unit magnitude point explosion source located at r = 0 m. The source wavelet is a Gaussian pulse with equivalent Ricker peak frequency of 30 Hz.

Figure 3. Time-domain Gaussian explosion filter responses, for ranges r = 5 m to r = 250 m. The thin red line represents the causal onset time of each filter. Beyond range r ~ 75 m, each filter is indistinguishable from a Gaussian pulse.

Figure 4. Pressure traces recorded at ranges r = 5 m to r = 250 m generated by a Gaussian explosion source of unit peak magnitude centered at r = 0 m. Trace amplitude (i.e., deflection) scale is identical to figure 2.

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EDITED REFERENCES

Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2011

SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for

each paper will achieve a high degree of linking to cited sources that appear on the Web.

REFERENCES

Aldridge, D. F., 2000, Radiation of elastic waves from point sources in a uniform whole space: Technical

Report SAND2000-1767, Sandia National Laboratories.

———, 2002, Elastic wave radiation from a pressurized spherical cavity: Technical Report SAND2002-

1882, Sandia National Laboratories.

Aki, K., and P. G. Richards, 1980, Quantitative seismology, theory and methods: W. H. Freeman and

Company.

Blair, D., 2007, A comparison of Heelan and exact solutions for seismic radiation from a short cylindrical

charge: Geophysics, 72, no. 2, E33–E41, doi:10.1190/1.2424543.

———, 2010, Seismic radiation from an explosive column: Geophysics, 75, no. 1, E55–E65,

doi:10.1190/1.3294860.

Heelan, P. A., 1953, Radiation from a cylindrical source of finite length: Geophysics, 18, 685–696,

doi:10.1190/1.1437923.

Jeffreys, H., 1931, On the cause of oscillatory movement in seismograms: Royal Astronomical Society,

Geophysical Supplement, 2, 407–416.

Poppe, G. P. M., and C. M. J. Wijers, 1990, More efficient computation of the complex error function:

ACM Transactions on Mathematical Software, 16, no. 1, 38–46, doi:10.1145/77626.77629.

Sharpe, J. A., 1942, The production of elastic waves by explosion pressures, Part I: Theory and empirical

field observations: Geophysics, 7, 144–154, doi:10.1190/1.1445002.

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