## 1 Introduction  
Ultrasound imaging systems transmit ultrasound into the body and reconstruct cross-sectional images based on the information received from the scattering wave. They are commonly used in clinical situations because they are applicable in real-time, are cost-effective compared to X-ray CT and MRI, and do not require a dedicated room. B-mode imaging is the most popular and widely used ultrasound-based imaging technique. This method uses the nature of wave interference to reconstruct a map of scattering intensity. The intensity values of B-mode images are relative values based on a spatial gradient of the acoustic impedance of the tissue. Therefore, B-mode is a qualitative imaging method, limiting its diagnostic potential [1; 2; 3; 4]. If implemented, quantitative imaging will help assess the efficacy of anticancer drugs on tumours over time and reduce operator-dependent inaccuracy in area and volume measurements [5; 6; 7]. In quantitative ultrasound imaging for the examination of breast cancer, sound speed imaging techniques using a ring array transducer have been investigated [8; 9; 10]. [PERSON] et al. [8] demonstrated the potential for detecting tumours in the breast by sound speed reconstruction based on transmitted waves, taking advantage of the fact that the tumour site is stiffer and has higher sound speed than other soft tissues. Subsequently, methods using-ray approximation [9; 10], diffraction tomography that considers refraction and diffraction [11; 12; 13], and full waveform inversion (FWI) that directly considers the wave equation have been proposed for higher-quality (high-resolution and low-artifact) images [14; 15; 16; 17; 18; 19; 20]. Currently, methods based on FWI are mainstream as high-quality reconstruction methods based on physics. On the other hand, FWI faces several challenges in its clinical application. One is the issue of computational cost: FWI is a method that optimises the sound speed distribution to create an observed signal in numerical simulations close to the real signal [14]. This optimisation process includes iterative calculations of the gradient method. To compute a gradient once, it is necessary to solve the forward problem of sound wave scattering twice. Therefore, the computational cost for convergence is significant and reconstruction cannot be carried out in real-time [16]. In addition, the problem is an inverse problem for a non-linear operator and has many local solutions [21]. Therefore, careful tuning is required in the design of the initial solution and regularisation parameters for implementation. These problems limit the clinical application of FWI. Meanwhile, deep neural networks (DNNs) have been investigated for medical imaging application in recent years [22; 23]. DNNs operate at high speed after training. Thus, the high-speed reconstruction of sound speed distribution using DNNs has been investigated. Several applications of DNNs in sound speed reconstruction have also been studied and it has been reported that sound speed distributions can be reconstructed from observed signals [24; 25; 26]. [PERSON] et al. [24] and [PERSON] et al. [25] reported that reconstruction with DNNs trained on datasets generated by numerical simulation can be of higher quality than those with FWI, and that the DNNs are robust against noise. In addition, [PERSON] et al. [26] showed that reconstruction is possible even by signals obtained from single linear array transducer. These studies have shown promise for practical applications of sound speed reconstruction using DNNs. The apparent performance of DNNs can vary significantly depending on how the training data and test data are chosen [27]. Therefore, DNNs require additional considerations compared to model-driven methods such as FWI. It can be expected that DNNs will show high performance when the training data contain the patterns of the test data. Conversely, there is a concern that performance may be significantly degraded when the training data do not contain the patterns of test data [28].