Temporal and Spatial Resolution and Aperture

In the introduction to this chapter, we suggested that our approach to separating Love wave modes would require that spatial and temporal data be transformed to frequency and wave number domain, and that the resulting K-f plot be used create dispersion curves. Thus, when designing a Love wave field experiment, the data must span the wave numbers and frequencies of interest to the investigator, and have sufficient resolving power to enable the investigator to distinguish between the frequencies and wave numbers of neighboring modes.

The temporal Nyquist criterion places an upper limit on the frequency that can be resolved with a given sampling rate:

∆tnyq = 1 2fmax

= π

ω (5.9)

where ∆tnyq is the Nyquist sampling interval, and fmax is the highest frequency be- ing measured. In practice, it is usually desirable to use sampling intervals that are considerably less than dictated by the Nyquist criterion. For the present discussion, we will use ∆t = 1/(5fmax) as an experimental design criterion. At the other end of the frequency spectrum, resolving low frequencies requires that data be collected over a duration comparable to a complete cycle of the lowest frequency of interest. For the present discussion, we will use a duration equal to twice the low frequency period as our design criteria. This duration, called the temporal aperture, is equal to (nt−1)∆t, so the minimum number of samples, nt, necessary to achieve a temporal aperture twice the longest wavelength being studied is:

Sources r01

∆r

receivers (geophones)

r₀₂

Figure 5.1: Source and receiver lay-out. The distance between a source and the first receiver, r0j, is called theoffset distance. ∆r is thereceiver spacing.

nt=

2

fmin∆t + 1 (5.10)

We denote the spatial wavelength of a signal using the Greek letterλ, so that the spatial wave number, 1/λ, represents the number of complete oscillations in a unit length. The spatial wavelength is related to the angular wave number, Kr by the relationship:

Kr =

2π

λ (5.11)

The angular wave number is expressed in terms of radians per unit length, while frequency is expressed as cycles per unit time. To simplify the following discussion, we will introduce a cyclic wave number, K =Kr/(2π), so that the Love wave phase velocity is: CL = ω Kr =f λ= f K (5.12)

A spatial analogue to the temporal Nyquist criterion is: ∆r= 1 2K = λ 2 = CL 2f (5.13)

where ∆r is the spatial sampling interval (receiver spacing). As was the case with ∆t, it is usually desirable to use sampling intervals that are considerably smaller than those dictated by the Nyquist criterion. We will use the following receiver spacing as an experimental design criterion:

∆r= 1 5K = λ 5 = CL 5f (5.14)

Spatial aperture is analogous to temporal aperture, and is equal to (nr −1)∆r. In order to resolve long wavelengths, we would like to have a receiver line of sufficient length (spatial aperture) to capture at least two wavelengths, so the minimum number of receivers, nr, will be:

nr =

2

K∆r + 1 =

2CL

f∆r + 1 (5.15)

In our discussion of the three-layer model, we examined dispersion and attenuation curves for frequencies as high as 120 Hz, which corresponds to a sampling interval of 0.00167 s. Although we explored the behavior of the system at frequencies as low as 4 Hz, our use of 10 Hz geophones obviates the need to design a data collection scheme for frequencies lower than 10 Hz. The minimum number of time samples (Equation 5.10) is 121. These sampling requirements are well within the capabilities of off-the-shelf hardware.

lay-out Vel (m/s) Low Frequency Limit (Hz) High Frequency Limit (Hz)

low den 100 2.1 50

low den 400 8.4 200

hi den 100 12.6 300

hi den 400 50.4 300∗

Table 5.2: Frequency ranges for the low density (1 m receiver spacing) and high density (0.167m receiver spacing) lay-outs. ∗ The high density lay-out upper limit is constrained by the temporal Nyquist frequency.

Receiver spacing corresponding to 120 Hz and 100 m/s is a very tight 0.167 m, and the number of receivers necessary to measure a 400 m/s phase velocity at 10 Hz is 480. Given the cost of receivers, we would like to reduce this number. Inspection of Figures 4.27 and 4.28 suggests that we could increase receiver spacing to 0.2 m, which corresponds to a frequency of 60 Hz and velocity of 100 m/s, and still see the important features of each curve, but this plan still requires 201 receivers.

More importantly, this exercise in fine tuning required such detailed a-priori knowl- edge of layer geometry and material properties as to obviate the need for an investiga- tion using Love waves. Realistically, collecting Love wave data may require more than one receiver lay-out. In this chapter, we will assume two separate receiver lay-outs, each using 96 receivers: A high density lay-out using 0.167m spacing (6 receivers per meter), and a low density lay-out using 1 m spacing. We will use these two lay-outs in this chapter’s simulations.

Throughout this discussion, we have implicitly assumed a flat, horizontally ho- mogeneous layer geometry. Although increasing aperture size improves resolution, it also increases the risk that the survey area will include discontinuities or asperities of sufficient magnitude to violate the assumption of horizontal homogeneity.