As a rock is compressed, the smallest aspect ratio voids are the first to be effectively closed. Though in reality the crack is never completely closed, it can become sufficiently stiff that the crack moduli approach the grain moduli of the rock. The stiffening of the cracks with increasing confining pressure causes P- and S-wave velocities in the rock to increase with confining pressure. This effect has been widely observed (e.g., Adams and Williamson, 1923; Birch, 1960; Christensen, 1982; Iturrino, 1995).
To determine the causes of the velocity variations within the central part of the aa flows and to separate the effect of the pore space from that of the solid rock on the seismic properties of basalt from Hole 990A, we inverted the high-pressure velocity data for the aspect ratio spectrum and apparent grain moduli for each sample listed in Table 1. The linearized form of the Kuster-Toksöz model, as described in Cheng and Toksöz (1979), was used in a constrained, Marquardt least-squares inversion of the velocity data.
The Kuster-Toksöz model does not specify which aspect ratios to use or how to choose which aspect ratios to model. The choice of aspect ratios is arbitrary, and the aspect ratio spectrum obtained by inverting the velocity data is thus nonunique because the aspect ratio spectrum depends on the initial choice of aspect ratios. We modeled the aspect ratio spectra of our samples in two steps. In the first step of the inversion, aspect ratios were chosen so that a specific crack type (as described by its aspect ratio) is effectively closed at each successive pressure increment. For example, the thinnest voids close at 10 MPa, the next set of voids close at 20 MPa, and so on. In addition to the calculated aspect ratios, aspect ratios of 1.0, 0.1, and 0.01 were included in the constrained inversion. A total of 14 different aspect ratios was initially used to model each sample.
The results of the first inversion demonstrated that aspect ratio spectra for the different samples were quite similar, but some calculated aspect ratios were not significantly different and some did not have significant concentrations. In the second step of the inversion, we chose just six aspect ratios that span the range used in the first inversion, 1.0, 0.1, 0.01, 0.005, 0.001, and 0.0005. The criterion for the number of aspect ratios used in the second inversion was that the RMS misfit for each sample modeled be <0.050 km s-1 (<2% RMS misfit). The results of this second constrained inversion are listed in Table 2, where the void concentrations are reported as porosities (in percent), which means the sum of the concentrations is equal to the porosity of the sample.
To compare the aspect ratio spectra of the samples, we used a normalized aspect ratio spectrum in which the normalized concentration is the total concentration divided by the porosity of the sample. The sum of the normalized aspect ratio concentrations is 1, and the porosity is a scaling factor. The normalized concentrations reveal a similarity in the aspect ratio spectra for the samples shown in Figure 3. Also shown are the averaged normalized concentrations for this suite of samples. The averaged normalized concentration represents the average aspect ratio spectrum for this suite of samples.
The similarity of the aspect ratio spectra for these samples (Fig. 3) suggests that the average of the normalized concentrations might apply to all of the samples (i.e., that the relative proportions of different cracks in all of the samples are essentially the same). If so, then the P- and S-wave velocities in any sample can be estimated from the porosity and the average aspect ratio spectrum with nearly the same accuracy and precision achieved by using the best fitting spectrum for that particular sample (Table 2). To test this hypothesis, we used the average spectrum to calculate the velocities in each sample and compared the calculated velocities with the observed velocities reported in Table 1. An example of the calculated velocities (for Sample 163-990A-10R-1, 12-14 cm) is shown in Figure 4. In all cases, the RMS misfit is <0.050 km s-1 (<2%). In two cases, the misfit is increased by a factor of two or three, relative to the misfit for the spectrum of that particular sample. In the remaining eight cases, the misfit is increased by <20%. To a very good approximation, these rocks can be described as having the aspect ratio spectrum that is summarized in Table 3.