## 1 Introduction  
The first quantitative ultrasound (QUS) technique for use in bone assessment was proposed by [PERSON] et al. [1] in the 1980s. Since then, a number of different QUS techniques have been proposed and developed [2; 3; 4]. QUS techniques were developed for early osteoporosis detection and fracture risk evaluation screening applications. When compared with X-rays, QUS offers several advantages: it is a noninvasive process and can be performed using portable and low-cost equipment. Techniques that can evaluate the quality of cortical long bone structures have recently attracted significant research attention [2; 3; 5; 6; 7; 8]. In this study, we have developed cortical long bone quality evaluation methods using the axial transmission (AT) technique [9; 10; 11; 12; 13]. A number of recent studies have used the AT technique, in which wide-band signals are emitted and the ultrasonic guided wave that propagates along the cortical bone is then analyzed. The cortical bone has previously been described as a transversely isotropic absorbing plate with finite thickness [14; 15; 16; 17; 18]. The ultrasonic guided wave that propagates in cortical bone consists of multiple propagation modes, and the frequency-dependent wavenumbers of these modes represent the elastic properties of the medium. Many techniques for frequency-wavenumber (_f-k_) analysis have therefore been proposed and reported. [PERSON] et al. proposed the Radon transform method, which estimates the phase velocity (\(c_{\rm p}\)) at each frequency [12]. The phase velocity is given by \(c_{\rm p}=2\pi f/k\), and the basic premises of the \(f\)-\(k\) and \(f\)-\(c_{\rm p}\) analysis methods are theoretically the same. The Radon transform method uses a single transmitter with multiple receivers, estimates the phase velocity using an iterative process, and produces high-resolution estimates. However, the computational cost of this method is not low because it requires inversion of a large-scale matrix. [PERSON] et al. [19] proposed a singular value decomposition (SVD)-based method and used a probe composed of multiple transmitters and emitters to analyze the guided waves. [PERSON] et al. extended this method and produced accurate depictions of the phase velocities of the plate [10]. Use of SVD allows the method to detect the modes with low intensity. The method was effective for estimation of the elastic properties of the medium [20]. However, the measurement resolution is determined by the aperture size, and it is necessary to estimate the number of propagation modes in the received signal. [PERSON] et al. proposed the sparse SVD (S-SVD) method, which combines the Radon transform method proposed by [PERSON] et al. with the SVD method [21]. The resulting method acquires super-resolution estimates using a combination of SVD and an iterative process. However, this method also requires estimation of the number of propagation modes in the received signal and multiple calculations for inversion of a large matrix. We recently developed an adaptive beamforming technique [22]. We used the estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm. This algorithm uses eigenvalue decomposition to separate the desired signal from noise. The ESPRIT algorithm also requires estimation of the number of propagation modes and the conventional method includes an iterative process with eigenvalue decomposition [22]. Because the matrix inversion and eigenvalue decomposition processes incur large computational costs, reduction of these computational costs would be useful for practical applications. Therefore, in this study, we propose a method that estimates the number of propagation modes with a low computational cost. The number of propagation modes in the received signal, which is denoted by \(M\) in this study, is usually estimated using a thresholding process. However, the eigenvalues that correspond to the signal are not equivalent to the signal intensities.