Recent seismological observations focusing on the collapse of an impulsive
wavelet revealed the existence of small-scale random heterogeneities in the earth medium.
The radiative transfer theory (RTT) is often used for the study of the propagation and
scattering of wavelet intensities, the mean square amplitude envelopes through random
media. For the statistical characterization of the power spectral density function (PSDF)
of the random fractional fluctuation of velocity inhomogeneities in a 3-D space, we use
an isotropic von Kármán-type function characterized by three parameters: the root
mean square (RMS) fractional velocity fluctuation, the characteristic length, and the
order of the modified Bessel function of the second kind, which leads to the power-law
decay of the PSDF at wavenumbers higher than the corner. We compile reported statistical
parameters of the lithosphere and the mantle based on various types of measurements for a
wide range of wavenumbers: photo-scan data of rock samples; acoustic well-log data; and
envelope analyses of cross-hole experiment seismograms, regional seismograms, and
teleseismic waves based on the RTT. Reported exponents of wavenumber are distributed
between

The first image of the solid earth is composed of spherical shells, for example,
PREM (preliminary reference Earth model)

When the central wavenumber of a wavelet increases much larger than the corner wavenumber
of the PSDF, the wavelet around the peak value is mostly composed of narrow-angle
scattering around the forward direction. In such a case, the Born approximation becomes
inappropriate; however, the phase shift modulation based on the parabolic approximation
is useful, which is called the phase screen approximation. As an extension of the RTT
with the phase screen approximation, the Markov approximation was also used for the
analysis of envelope broadening and peak delay with increasing travel distance

There have been various kinds of measurements of the PSDF of the random velocity fluctuation, where the PSDF is often supposed to be a von Kármán type. In the following section, the main objective is to compile reported PSDF measurements in various scales in different geological environments of the solid earth: photo scanning of small rock samples, acoustic well logs, array analyses of teleseismic waves, waveform analyses using finite difference (FD) simulations, and analyses of seismogram envelopes on the basis of the RTT. We enumerate their statistical parameters and plot their PSDFs against wavenumber. We will show that the envelope of all the PSDFs is well approximated by a power-law decay curve. Then, we will discuss the results obtained and a few problems in the envelope synthesis theory for such random media and the geophysical origin of such power spectra.

We consider the propagation of scalar waves as a simple model, where the
inhomogeneous velocity is given by

There are several types of PSDF and ACF characterized by a few parameters.

The ACF is written by using a modified Bessel
function of the second kind of order

Especially for an anisotropic case, we define the von Kármán-type PSDF in 3-D

For a case formally
corresponding to

Gaussian-type ACF and PSDF are also used because they are mathematically tractable.

We plot those PSDFs against wavenumber and ACFs against lag distance in Fig.

There are several kinds of measurements for estimating statistical parameters
characterizing random media. Here we principally collect measurements supposing a von
Kármán-type function for isotropic randomness; however, we include a few measurements
supposing anisotropic randomness and a Gaussian-type function. On a small scale, the photo-scan method is
applied to small rock samples. Acoustic well logs are available in deep wells drilled in
the shallow crust. When the precise velocity tomography result is available, we can
directly calculate the PSDF. In seismology, the most conventional method is to analyze
seismograms of natural earthquakes or artificial explosions after traveling through the
earth heterogeneity. It is better to focus on MS amplitude envelopes (intensity time
traces) since phases are complex and caused by random heterogeneities. Comparing observed
seismogram envelopes with envelopes synthesized in random media, we can evaluate von
Kármán parameters. For the synthesis, we can use the FD simulations, the RTT with the
Born approximation, and the RTT with the phase screen approximation that is equivalent to
the Markov approximation. For each reported measurement, we enumerate the target region,
data and the method, the measured PSDF as a function of wavenumber

Reported von Kaŕmán parameters of rock samples and acoustic well logs. A
value in parentheses ( ) is a priori assumed in the measurement. A label with an asterisk

The photo-scan method uses a scanner to take a picture of the polished flat surface of a
small rock sample

In the case of isotropic randomness, we evaluate the 1-D PSDF from the 3-D ACF along the

Supposing the randomness is isotropic, we evaluate corresponding 3-D PSDFs of
R1–R5 and plot them in Fig.

An acoustic well log is obtained from the measurement of the travel time of an ultrasonic
pulse along the wall of a borehole. Measurements W1 (volcanic tuff) and W2 (tertiary to
pre-tertiary) in Japan clearly show power-law decay with

We note that

Reported von Kaŕmán parameters of the lithosphere including the crust and
the uppermost mantle. A value in parentheses ( ) is a priori assumed in the measurement.
When the estimated value is given by a range, a value in square brackets [ ] is used for plotting the PSDF. A label with an asterisk

There have been measurements of velocity tomography at various scales, from which we can
calculate the PSDF and then estimate von Kármán-type parameters. This method depends
on the spatial resolution of the tomography result. Measurement L1 in Table

Reported von Kaŕmán parameters of the upper mantle and the lower mantle. A
value in parentheses ( ) is a priori assumed in the measurement. When the estimated value
is given by a range, a value in square brackets [ ] is used for
plotting the PSDF. A label with an asterisk

3-D PSDF vs. wavenumber for

Teleseismic

3-D PSDF vs. wavenumber for the upper and lower mantle. See labels
in Table

FD simulation is often used for the numerical simulation of waves in an inhomogeneous
velocity structure. For the evaluation of average MS amplitude envelopes, we have to
repeat simulations of the wave propagation through random media having the same PSDFs
that are generated by using different random seeds. There are several measurements of
statistical parameters using FD such as L9–L11 and ML4 in Tables

3-D PSDF vs. wavenumber for all the data.

The RTT is essentially stochastic to directly synthesize the intensity (the average MS amplitude envelope) of a wavelet propagating through random media. There are two conventional methods on the basis of the RTT: one uses the Born approximation and the other uses the phase screen approximation based on the parabolic approximation when the wavenumber is larger than the corner. The former neglects the phase shift, but the latter correctly considers the phase shift.

We here study the deterministic scattering of scalar waves by a single spherical obstacle
(radius

Interpreting

Deterministic scattering of scalar waves by a high-velocity sphere.

For uniformly distributed random media characterized by

Plot of

In the framework of the RTT, the Monte Carlo simulation is a simple method to
stochastically synthesize the wavelet intensity time trace. A particle carrying unit
intensity is shot randomly from a point source, and its trajectory is traced with the
increment of time steps. The occurrence of scattering is stochastically tested by the
inequality

Flowchart of the Monte Carlo simulation code according to the RTT
for the scalar wavelet intensity in uniform random media.

The RTT for the scalar wave case can be extended to the elastic vector wave case by using
Stokes parameters. There are four scattering modes: PP, PS, SS, and SP scatterings, and
the

The RTT with the Born approximation has been often used not only for the analyses of

Most measurements of

When

When this approximation is used,

3-D PSDF vs. wavenumber for the crust and the upper mantle. Data of Gaussian-type, anisotropy type, strong heterogeneity, the lower mantle, and the whole mantle are excluded. The light-gray straight line visually fits to most spectral envelopes.

Some measurements a priori assumed

Plotting PSDFs against wavenumber is more informative for understanding the random
heterogeneity compared with enumerating statistical parameter values.
Figure

Eliminating data supposing a Gaussian-type data LA1–LA3, strong heterogeneity data
LS1–LS4, anisotropy-type data L1 and LS5, and the lower mantle and the whole mantle data
ML1–ML4 and MW1–MW2 from Fig.

We draw a power-law decay line PSDF

It will be necessary for us to measure the small-scale randomness of
sedimentary rock samples. More measurements are necessary in the wavenumber
range

In most PSDF measurements, each power-law decay branch is short since the Born
approximation senses the spectrum up to twice the wavenumber. It will be necessary to
measure how each power-law decay branch varies with wavenumber increasing. It will be
necessary to estimate the corner

Although most measurements used in this review analyzed intrinsic attenuation, we did not enumerate them in this review since different assumptions were used in different measurements. It will be necessary for us to systematically measure the PSDF of random heterogeneity in conjunction with intrinsic attenuation.

We should note that there are large variations in

Figure

In Sect. 3.6.1, we mentioned that the conventional Born approximation is
inapplicable and the phase screen approximation is useful when the phase shift becomes
large as the wavenumber increases. In order to avoid the difficulty, taking the center
wavenumber of a wavelet as a reference,

We note that some papers numerically show that the RTT with the Born approximation works
well in some cases over the above limitation.

Spectrum division method.

Acoustic well-log and photo-scan methods faithfully measure the inhomogeneous elastic coefficients. The RTT used here supposes the scattering contribution of random inhomogeneity of elastic coefficients only; however, observed seismograms do not only reflect those but also the scattering contribution of pores and cracks distributed over the earth medium. It will be necessary for us to study their contribution in the intensity synthesis.

In observation, we may take the power-law spectral envelope as a reference curve for
studying the regional differences, especially in the power-law decay part of the PSDF.
The characteristic length

The power-law decay spectral envelope reminds us of the observed fractal nature of
various kinds of surface topographies:

For igneous rocks such as granite, there are variations in composition of minerals and
grain sizes, which depend on a variety of slow crystallization differentiations of
basaltic magma. Random variations in acoustic well-log profiles reflect the complex
sedimentation process during the geological history. Volcanism produces more
heterogeneous structures composed of pyroclastic material and lava. For random
heterogeneities in the mantle, we imagine complex mantle flow.

In advance of the measurements based on the RTT for anisotropic scattering presented
here, there have been many measurements of the isotropic scattering coefficient

Recent seismological observations focusing on the collapse of an impulsive wavelet
revealed the existence of small-scale random heterogeneities in the earth medium. The RTT
has often been used for the study of the propagation of wavelet intensities, the MS
amplitude envelopes. For the statistical characterization of the PSDF of random velocity
inhomogeneities in a 3-D space, we have used von Kármán type functions with three
parameters: the RMS fractional velocity fluctuation

All the data for this review work are listed in Tables 1–3, where all the references are given.

The author HS reviewed reported measurement of statistical parameters characterizing the random heterogeneities in the earth.

The author declares that they have no conflict of interest.

This paper is a part of the Beno Gutenberg medal lecture at the 2018 EGU assembly held in Vienna. The author is grateful to the seismology section of EGU for giving him an opportunity for reviewing measurements of random heterogeneities in the solid earth and various theoretical approaches. The author expresses sincere thanks to his colleagues and ex-graduate students who enthusiastically collaborated with him in studying seismic wave scattering in random media. The author acknowledges reviewers, Vernon Cormier, Michael Korn, and Ludovic Margerin and editor Tarje Nissen-Meyer for their helpful comments and suggestions. Edited by: Tarje Nissen-Meyer Reviewed by: Ludovic Margerin, Michael Korn, and Vernon Cormier