NONLINEAR ULTRASOUND IMAGING ∗ SEBASTIAN ACOSTA † , GUNTHER UHLMANN ‡ , AND JIAN ZHAI § Abstract. We consider the ultrasound imaging problem governed by a nonlinear wave equation of Westervelt type with variable wave speed. We show that the coefficient of nonlinearity can be recovered uniquely from knowledge of the Dirichlet-to-Neumann map. Our proof is based on a second order linearization and the use of Gaussian beam solutions to reduce the problem to the inversion of a weighted geodesic ray transform. We propose an inversion algorithm and report the results of a numerical implementation to solve the nonlinear ultrasound imaging problem in a transmission setting in the frequency domain. Key words. ultrasound imaging, nonlinear acoustic waves AMS subject classifications. 35R30, 92C55 1. Introduction. Nonlinear ultrasound waves play an increasingly important role in diagnostic and therapeutic medicine. Improvements are being accomplished for the ultrasonic imaging of blood perfusion in organs and tumors [46, 52, 17, 44, 14, 5, 28, 29], high-intensity focused ultrasound ablation of pathological tissues [45, 13, 33], and drug/gene delivery using micro/nano agent assisted ultrasound [10, 11, 12]. We are also motivated by the portability of ultrasound-based technologies which makes them ideal for monitoring patients in the operating room. For instance, there is an alarmingly high incidence of brain and kidney damage in neonates who undergo open- heart surgery to palliate congenital heart defects. Depending on the type of cardiac surgery, several studies have documented an incidence of organ injury ranging from 35% to 75% [2, 21, 27]. This problem reveals an unresolved need for better and portable imaging techniques to monitor perfusion during surgical procedures. The complexity of physiological media surrounding blood vessels limits the ap- plication of ultrasound to assess perfusion status. In order to reduce the influence of media heterogeneity and enhance the visualization of blood vessels, special mi- crobubble contrast agents have been designed [52, 17, 44, 42, 14, 5, 18, 17, 20]. Once injected in the blood stream, these enhancing agents induce a nonlinear response upon interaction with ultrasound waves. The nonlinearity generates vibration frequencies different from the isonating frequency. This effect can be measured and processed to form images of the source of nonlinearity while simultaneously filtering out some of the confounding interaction with the heterogeneous media. The main goal of this paper is to contribute to the mathematical understanding of quantitative nonlinear ultrasound imaging. Our objective is to determine whether boundary measurements of the ultrasound field can uniquely determine the coefficient of nonlinearity in the wave equation. Our starting point is a lossless nonlinear wave equation of Westervelt type [44, 17, 18, 38] governing the propagation of waves in a bounded domain Ω ⊂ R 3 with smooth boundary ∂ Ω. This model has the following * Submitted to the editors DATE. † Department of Pediatrics-Cardiology, Baylor College of Medicine and Texas Children’s Hospital, Texas, USA (sebastian.acosta@bcm.edu) ‡ Department of Mathematics, University of Washington, Seattle, WA 98195, USA; Institute for Advanced Study, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China (gunther@math.washington.edu). § Institute for Advanced Study, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China (jian.zhai@outlook.com) 1 arXiv:2105.05423v2 [math.AP] 6 Jul 2021