Introduction
The dispersion associated with flexural-wave propagation in floating ice sheets is related to the elastic properties and thickness of the ice, and measured dispersion may be inverted to yield the elastic properties and ice thickness. The use of dispersion measurements in this manner has been summarized by Reference Anderson and KingeryAnderson (1963) and the measurements have received fairly wide application (Reference Clements, Clements, Willis and WilsonClements and others, 1958; Reference CraryCrary, 1954; Reference Oliver, Oliver, Crary and CotellOliver and others, 1954; Reference Press and EwingPress and Ewing, 1951; Reference RichterRichter, unpublished).
The conventional practice used for the determination of flexural-wave dispersion is to establish two recording sites and determine the frequency-dependent phase shifts occurring as a result of propagation between the sites. The end result is the determination of average elastic properties and ice thickness along the path separating the recording sites. While yielding satisfactory results, this procedure requires the transport of equipment over the ice sheet and the presence of both personnel and equipment in the area of interest. In addition, the study of azimuth dependence in the area of interest requires the movement of at least one recording site or the presence of more than two sites.
A method proposed by Reference Alexander and TaylorAlexander and Taylor (1969) and employed by Reference TaylorTaylor (1972) for the study of Rayleigh-wave propagation, provides an alternative to the conventional method of determining flexural-wave dispersion. A version of this procedure modified for use in ice studies is presented here.
Theory
The frequency-domain representation of a propagating flexural wave may be written as,
where the transfer function of the recording system
the Fourier transform of the source excitation function
the transfer function of the ice sheet
k being the wave number, α the material attenuation of the ice, and x the distance from the source. Thus, for the geometry shown in Figure 1, the ratio of R 2(ω) to R 1, (ω), assuming identical sources, may be written as,
Fig. 1. The source–sensor geometry required for single-sensor measurements.
Equating the phase spectra and solving for the phase velocity yields,
where C(ω) is the phase velocity for propagation across path Δx and Ψ(ω) is calculated from the ratio R 2(ω)/R 1(ω). The term 2Nπ represents the ambiguity present in all phase spectra. Equating the amplitude spectra and solving for the attenuation yields,
and |A(ω)| is known on the basis of the ratio R 2(ω)/R 1(ω).
Thus, the phase velocity and attenuation appropriate to the path △x, may be determined from a single phone measurement made colinear with but remote from the path. The results are independent of propagation effects along the common path x 1, and the response of the recording system I(ω), and no assumptions of uniform ice thickness or elastic properties are necessary. The only necessary assumptions are that the sources are identical to within the desired accuracy of the measurements. It should also be noted that it is not necessary to know the sensor-source distances x, or x 2; only the source separation △x is required.
The assumption of identical source excitation functions is reasonable for any underwater explosive source. However, even this assumption would not be necessary if another sensor were available at an arbitrary distance on the opposite side of the area of interest. The response of this sensor would again cancel and there would be no need to match the characteristics of the sensors. The use of this second sensor would also eliminate any need for origin time of the source.
Results and conclusions
The single-phone measurements described here were taken on Pike Lake, Washington County, Wisconsin, U.S.A., and on Green Bay, 10 km east of Marinette, Wisconsin, U.S.A. The sources in both studies were blasting caps detonated beneath the ice. A typical recording obtained at each of the sites is shown in Figure 2. The flexural wave is easily identified by its amplitude and inverse dispersion. There is little indication of any contamination by other modes and both recordings are fairly broad-band. The lack of contamination and broad-band nature of the flexural waves generated by sources located under the ice is expected on the basis of the results of Reference Ewing, Ewing, Jardetzky and PressEwing and others (1957, p. 306).
Fig. 2. Typical recordings resulting from the underwater detonation of blasting caps at A, the Pike Lake site and B, the Green Bay site.
The measured phase velocities and attenuations for each site, calculated on the basis of the procedure described here, are shown in Figures 3 and Figures 4respectively. To demonstrate the validity of the single-phone phase velocities the data of Figure 3 were used to derive the thickness and Young’s modulus appropriate to the ice at each site. This was accomplished by assuming a Young’s modulus of 5 X 1010 dyn cm–2 (5 X 109 Nm–2) and then calculating theoretical phase velocities in the pass band from 1 Hz to 50 Hz for ice thicknesses from 0.1 in to 1 m in 1 cm increments. The thickness of the model for which calculated phase velocities best represented the measured phase velocities was then assumed to be the ice thickness. Values of Young’s modulus for this model were then varied from the original value to improve the correlation between measured and calculated velocities. The final results of this procedure are summarized in Table I.
Table I. Ice thickness, Young’s modules, and attenuation values
The results shown in Figures 3 and Figures 4and Table I establish the validity of the single-sensor method for flexural-wave dispersion measurements. The procedure as used required at most one individual in the area of interest and, thus, minimizes the problems associated with work on ice sheets. Alternately, no personnel would be required in the area of interest if the sources were projected into the area from some outside point. The procedure also minimizes the required instrumentation, with consequent economic and logistical benefits, and eliminates the need for matched sensors.
Fig. 3. Flexural-wave dispersion as determined from single-sensor recordings.
Fig. 4. Flexural-wave attenuation as determined from single-sensor recordings.
Acknowledgement
The effort of Mike Richter, Department of Geological Sciences, University of Wisconsin–Milwaukee, in conducting the field measurements was greatly appreciated. The research which employed the single-sensor dispersion measurements was supported by National Science Foundation Grant No. KG-41011 for the study of elastic properties of fresh-water ice.
