## 1 Introduction  
Doppler ultrasound is widely used to estimate velocity of blood flow and tissue motion in biological specimens [1]. Traditional Doppler approaches estimate the axial component of motion, where the maximum velocity is limited by the pulse-repetition frequency (PRF) of the ultrasound transmission as described by the Doppler equation (see Equation (9) in Section 2.1.5). In the most common estimation algorithms, such as the lag-one autocorrelation approach [2], velocities that exceed this limit result in phase wrapping (i.e., aliasing) and can result in estimates in the opposite direction of the actual flow. However, using conventional ultrasonic methods, if the Doppler estimation occurs throughout an image frame, such as in color flow Doppler, then the maximum velocity is now limited by the frame rate. With conventional ultrasonic methods, images are formed by using one focused transmit per lateral scan line. This design decision, while simple to implement and well understood, results in a temporal bottleneck on the frame rate. Because the total acquisition time for each transmit event is the round-trip time of flight to the maximum depth of interest and back, the total image acquisition time is scaled by the number of transmit events (on the order of 128 or more). Thus, if high frame rate images are desired, reducing the number of transmit events for each frame is paramount. Coherent compounding approaches were introduced to minimize the number of transmit events for each image frame and thereby increase the absolute frame rate without sacrificing image quality. These approaches include limited diffraction beams [3; 4; 5], steered plane waves [6], and diverging waves [7; 8]. Each of these methods produces high-quality images at frame rates in the kHz range. Acquiring data at the highest frame rate is paramount for accurate reconstruction of high velocity motion, such as that observed in mouse blood flow, where velocities can be as high as 900 mm/s [9]. This is particularly true for high-frequency ultrasound (>20 MHz) because, based on the Doppler equation, an increase in transmit frequency will lower the maximum velocity that can be resolved before aliasing. Compounding approaches result in an effective slow-time downsampling of the data, thus reducing the frame rate by a factor of \(N_{\phi}\), where \(N_{\phi}\) is the number of angles transmitted per compounded frame. In addition, because the plane-wave compounding approach relies on a stationary medium assumption, the compounding process can result in signal loss when particles move a half-integer multiple of a wavelength over the compounding interval [10]. For this reason, using the precompounded data is advantageous for high velocity imaging. This idea has been successfully exploited to compensate for motion artifacts in coherent compounding techniques [11; 12]. Use of the precompounded data for Doppler estimation comes with its own costs. Because each precompounded image is transmitted at a different steering angle, sidelobes in the point spread function (PSF) rotate about the target location even when the target is stationary. This sidelobe motion results in a periodic noise source which can corrupt velocity estimates. In order to address this issue, we propose using a repeated transmission sequence to avoid sidelobe motion on a short time scale. Specifically, we propose two new approaches which use this new transmit sequence. The first method estimates flow by evaluating the slow-time autocorrelation between adjacent precompounded images at the same transmit steering angle and aggregating the individual autocorrelation estimates over all angles. The second approach generates compounded frames for the initial and repeat events, respectively, and correlates these two frames. For both approaches, the autocorrelation occurs at a lag equivalent to one transmit event as opposed to a full compounded frame, resulting in an increase of the Nyquist velocity on the order of the number of angles used in the transmit sequence.