## 1 Introduction
This method employs a small transducer and an ultrasonic pulse with a frequency spectrum higher than 0.5 MHz, a narrow diffusion angle and high resolution according to the equation for the beam spread angle [29]. In this method, a two-channel ultrasonic transducer playing the role of both transmitter and receiver is installed at the exposed end of the steel rod and the time taken by the ultrasonic pulse reflected from the buried end of the rod to re-emerge is measured. If the traveling velocity of the ultrasonic pulse in steel is exactly known, the embedded depth of the rod can be estimated by means of the measured time. However, concrete is the most widely used construction material in the civil engineering field, and has non-homogeneous characteristics, so when high-frequency ultrasonic waves are used, attenuation such as scattering occurs severely. Therefore, these methods have a disadvantage in that an ultrasonic device that is less useful in the civil engineering field must be used. Because of the narrow diffusion angle exhibited by ultrasonic frequency spectra higher than 0.5 MHz, the ultrasonic pulse must be emitted in the axial direction of the rod and the structure installed above the exposed end of the steel rod must be completely removed. In addition, if the embedded end of the rod is broken and presents irregularities, the reflected wave will not propagate axially and will likely prevent consistent measurement [30; 31; 32]. In a will to settle such drawbacks, this study proposes a method estimating the embedment depth of anchor bolts by using ultrasonic P-waves with frequency range of 50-100 kHz that are commonly used for the NDT of concrete. In the P-waves, the oscillations occur in the direction of wave propagation. P-waves are less sensitive to the sample-transducer contact problem than S-waves and allow the more accurate velocity measurements in concrete where higher signal-to-noise ratios can occur [33]. The utilization of a basic ultrasonic device measuring the transmitting-receiving time gap is devised to reduce the economic burden by using highly practicable equipment in the concrete NDT of civil structures. Furthermore, the ultrasonic P-wave for concrete is featured by wide spreading, which allows paying no regard to the reflection angle nor the transmission angle. The proposed method uses ultrasonic P-waves with a frequency spectrum of 50-100 kHz that is capable of penetrating concrete and steel and an ultrasonic device measuring the time gap between the transmission and the reception of the pulse. The moving path of the P-wave is analyzed using the spreading property of the P-wave with a frequency range of 50-100 kHz and a formula was established to obtain the transmission-reception time gap according to the embedment depth of the anchor bolt. The defined formula is solved by inverse analysis using equation error estimation (EEE) to estimate the embedded length of the anchor bolt. The instability generated in this process is stabilized by regularization. The proposed method is validated by a numerical example and an experiment. The numerical example simulates an anchor bolt installed in a concrete member with a rectangular section and infinite length and the analysis is conducted considering the measurement errors as well as the errors in the major variables. In addition, the results were analyzed with respect to the changes in the regularization factor to propose a guideline for determining the appropriate value of the regularization factor since the factor plays crucial role in controlling the stability and accuracy of the EEE with the regularization. The experiment performs ultrasonic measurement on prismatic concrete specimens in which anchor bolts were installed. Both the numerical example and experiment demonstrated the high accuracy of the proposed method in estimating the embedment depth of the anchor bolt.