1-D inversion

The simplest inversion that we can possibly do is assuming that the medium is homogeneous isotropic. What we obtain is the mean velocity (equation (7)), in this case V_iso=8452 ft/sec. The next step is to assume that the model is still homogeneous but elliptically anisotropic. Using equation (8) we find that V_x = 8586 ft/sec and V_z = 8079 ft/sec. Notice that for this particular recording geometry (Figure 9), V_iso is closer to V_x than to V_z, which means that the ``averaging'' of the horizontal and vertical directions that the isotropic inversion implicitly does is not a simple arithmetic average. When the model is also heterogeneous, the same conclusion may be drawn as it will be show later.

To measure the goodness of the fit between measured and calculated traveltimes, we use the mean-absolute value of the mismatch

$$error \ =\ \frac{1}{M} \sum_{i=1}^{M} \vert t_{c_{i}} - t_{r_{i}} \vert,$$ (12)

where t_r<<382>>i and t_c<<383>>i are the measured and calculated traveltimes respectively and M is the total number of traveltimes.

When the estimated V_iso in the homogeneous model is used to compute synthetic traveltimes, error= 1.04 ms. When the model is homogeneous anisotropic, error= 0.94 ms.

The result of the isotropic inversion assuming a layered medium is shown in Figure . Only traveltimes corresponding to rays below 2705 ft and above 4000 ft were used. This depth interval was discretized in 60 horizontal layers of equal thickness (21.583 ft). Straight rays were used to compute synthetic traveltimes since small velocity variations are expected in this site (Harris et al., 1990b). Conjugate gradients iterations (40) were performed until no appreciable changes were seen neither in the model nor in the mean-absolute value of the error (equation (12)). This corresponds to reaching the flat part of the hills in Figure (5). For the model shown in Figure , error = 0.67 ms.

 

Isos
Figure 11 Result of the isotropic layered inversion. The thickness of each layer is 21.583 ft

Now, we allow the model to be anisotropic. The result of the inversion is shown in Figure . For traveltimes computed through this model, error=0.59 ms. The thick curve represents the horizontal velocity and the thin one represents the vertical velocity. The first thing we notice is that as expected V_x is generally larger than V_z. Figure  compares V_x and V_z with V_iso. In general, V_iso is closer to V_x than it is to V_z, which is consistent with the previous the results of the inversion assuming an homogeneous medium. This means that for the type of recording geometry used (ray angles between and $\pm 50$ degrees) the isotropic inversion is affected primarily by the horizontal component of the velocity. Since there are fewer rays at large angles, the isotropic inversion is less constrained by them. However, rays at large angles contain independent information that might be important to improve horizontal resolution in 2-D models.

 

Anisxsz
Figure 12 Result of the anisotropic layered inversion. The thickness of each layer in 21.583 ft The differences between V_x (thick line) and V_z (thin line) are represented by two colors: light gray, when V_x > V_z and dark gray when V_x < V_z.

Sonic logs were available in this site at both wells (Figure ). They sample the vertical velocity about the well at frequencies ($\sim$ 10 kHz) much larger than the typical frequency of the cross-well data ($\sim$ 1 kHz). To compare the information obtained from this two types of measurements (1-D tomogram and velocity logs), we did some averaging to the logs. First, we averaged each slowness log in blocks of equal thickness and equal to the layer thickness in the 1-D tomographic inversion. Secondly, the two averaged slowness logs were averaged again into a single one. The purpose of the last averaging was to simulate the horizontal averaging that the 1-D tomographic inversion implicitly does. Figure  compares the average velocity log with V_iso, V_x and V_z. Note that V_z is not only much closer to the average velocity log (as expected) but also better correlated with it, when compared with V_iso and V_x.

In the anisotropic inversion, we found that for this particular data set 60 layers of 21.583 ft each was a good compromise between resolution and stability. Reducing the layer thickness by half has the effect (not shown) of increasing the resolution at the expense of large variations and instabilities in the vertical component of the velocity

 

IsosAnisxsz
Figure 13 Comparison between isotropic and anisotropic layered inversion. Left: V_iso (thin line) and and V_x (thick line). Right: V_iso (thick line) and V_z (thin line). The differences between the isotropic and anisotropic inversion are represented by two colors: light gray when V_iso > V_x or V_z, and dark gray when V_iso < V_x or V_z.

that is not well sampled by the recording geometry. The horizontal component of velocity is generally more stable than the vertical for smaller layer thicknesses. Obviously, increasing the layer thickness made the inversion more stable at the expense of less resolution.