We present general formulation that can compose various methods to determine phase velocities, arriving directions of Rayleigh and Love waves, and particle motions of Rayleigh waves, by using a circular array data set of three-component microtremors. We also present formulas to examine the effects of both the number of sensors consisting of a seismic array and the presence of noise on the analysis results. In the formulation, we postulate that a wavefield of microtremors is a set of plane waves propagating in a 2D plane, polarized in the horizontal directions, and that each wave is not correlated with others. Our formulation is valid under general situations where plane waves arrive from different directions with different intensities.
The data processing procedure to apply our formulation is as follows. Horizontal-component waveforms provided from a circular array are compiled into two kinds of data sets of waveforms: radial and tangential components with respect to the circular array. Respective component waveforms are Fourier-expanded with regard to the angular-axis at each time, so that the resulting Fourier coefficients form complex time histories. Those time histories are used to estimate either cross or power spectral densities, from which we extract the information of interests, such as phase velocities, arriving directions, and particle motions. The formulations presented in this study describe the relation between the spectral densities and those quantities.

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The spatial autocorrelation (SPAC) method requires a special circular array where several observation points are equally spaced on the circumference. In order to look for the possibility of developping a new method of array observation with fewer restriction of arrangement than the SPAC method, we proposed a formula of the complex coherence function (CCF) of the Rayleigh wave measured on a couple of observation points located at any place. This formula was derived on the basis of an analytical solution of Lamb's problem, aiming to study the relation between wave source and observation point. The formula was given as simple discrete representation consisting of the Bessel function of the first kind of zero order J₀ (kr) (k: wavenumber, r: radius of array) and an infinite series with higher-order Bessel functions. In the SPAC method, by regarding the SPAC coefficient from directional average of CCFs (real part) as J₀ (kr), phase velocities of Rayleigh waves (wave number k) are calculated.
We first studied the relationship between the values of CCF and wave sources located far from a couple of observation points, and found that the values of CCF strongly varies depending on the direction with increase in kr, and also found that such directional properties were mainly caused by the variation of the infinite series in the formula of CCF. Furthermore, we applied the formula to the SPAC method for revealing the mechanism of the directional average and the reason why the SPAC method requires the special circular array with sensors equally spaced on a circle. The results are summarized as follows: 1) The values of the infinite series gets lower enough to be negligible after the directional average of CCFs, so that the SPAC coefficient can be approximated to J₀ (kr). 2) From the inverse analysis on the condition that the values of the infinite series is equal to zero, it was found that the condition was satisfied not only in the usual SPAC arrays but also in some extra circular arrays consisting of observation points not equally spaced on the circumference.
This result suggests the possibility of array design with fewer restriction of arrangement of observation points, using a new algorithm for obtaining J₀ (kr) without the operation of directional average used in the SPAC method.

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The surveys for the tunnels at depths of about 15 m and 2 m were made with a high frequency controlled source magneto telluric (HFCSMT) system, of which transmitter is a horizontal loop-coil, in Kouyama and Sueyoshi, Kagoshima prefecture, respectively. These tunnels in Shirasu layer were the air-raid shelters made in the Second World War. In Sueyoshi, the survey was performed for three positions of the transmitter that are at the distance of 20, 40 and 100 m from the receiver.
The changes of the apparent resistivity values related to the cavity are found clearly just upon the tunnels. These changes are intensively larger than those calculated in the 2-dimensional resistivity model based on the MT theory. In Kouyama, a source-receiver distance is so short that the far-field condition is not satisfied, because of the geometry of the survey area and low power of our transmitter. The result in Sueyoshi implies that the apparent resistivity values for underground cavities are strongly amplified in the near-field region for electromagnetic waves from a vertical magnetic dipole.
In order to examine this idea, the experiment was conducted on the farm of Saga University; a cavity of one meter in diameter and of 1.3 m in length was buried at a depth of 2.3 m. The observations were made also for various transmitter-receiver distances. The experiment intensely supports the results of the surveys, because the phenomena that the apparent resistivity contrast increases as decrease the source-receiver distance is clearly seen. Therefore, the HFCSMT method with the horizontal loop-antenna in the near-field condition is effective to detect underground cavities and tunnels and to estimate their horizontal positions at the shallow part of the subsurface.

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Precise measurement of drill head position is required to avoid the collision between the drill head and underground facilities, when a horizontal drilling machine is used for constructing underground gas pipes in the city. A positioning technique without special devices on the ground surface has been expected especially beneath the waterway or heavy traffic road, because it is difficult to conduct the measurement for locating the drill head position on the ground surface.
The authors have studied both magnetic and pitch sensors installed inside the drilling tool for locating drill head position, and we have found that alignment of two sets of magnetic sensors is effective to reduce noise in the city which disturbs the natural magnetic field. We have confirmed that the sensors can keep the drill head position with a precision less than 10 cm, when the drill head progresses 12 m along the drill hole. Engineers and workers do not have to walk around just above the underground drilling site to locate the drill head position by using the sensors.

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Nettleton's method is a method to select the most appropriate Bouguer reduction density minimizing the correlation between Bouguer gravity anomalies and topographic elevations. In this method, a line profile of gravity values is taken over a topographic feature not associated with density variations or subsurface structure. The authors reconsidered mathematically this method with an expansion from a line profile to 2-D values. The obtained results show that the density ρ_a by Nettleton's method, with non-correlation between gravity values and elevations, corresponds to the most appropriate density ρ_r by Rikitake et al.'s method.
Bouguer anomalies in the eastern part of Yamaguchi prefecture with the density ρ_a reflect the distribution of strata composed of the Cretaceous metamorphic rocks and Jurassic sedimentary rocks.

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