## 1 Introduction  
Spatial coherence (SC) of ultrasound backscattered echoes has been the object of numerous works in the ultrasound imaging field. The first works date back to the 90s [1; 2; 3; 4; 5], when it was proposed to extend to pulse-echo ultrasound the Van Cittert Zernike (VCZ) theorem of statistical optics, which describes the spatial covariance of the wave field generated by an incoherent source [1]. Recently, the attention has been focused towards exploiting the concept of spatial coherence for the development of new image reconstruction and beamforming techniques. Some of the most renowned coherence-based methods are coherence-factor [6], phase-coherence and sign-coherence [7] beamforming, Filtered-Delay Multiply and Sum (F-DMAS) beamforming [8], and Short-Lag Spatial Coherence (SLSC) imaging [9]. The latter is a technique which directly generates an image of the spatial coherence of backscattered echoes evaluated at short lags, while all other methods aim at generating B-mode images, either via a coherence-based weighting or through non-linearcorrelation-like operations as in F-DMAS. Anyway, all these techniques use some estimate of spatial coherence to increase image quality. In general, they show enhanced clutter rejection and contrast improvement capabilities [9; 10], which often also come along with an increase of lateral resolution [7; 8]. For these reasons, in several studies, they have been proposed for use in application to techniques that partially sacrifice some of the above-mentioned image quality metrics to improve other crucial factors in ultrasound medical imaging, such as the frame rate. This is the case of well-known techniques like plane-wave imaging [11; 12], multi-line acquisition (also called parallel beamforming) [13], and multi-line transmission (MLT) [14]. MLT, as the name suggests, is based on the use of multiple beams, which are transmitted simultaneously in the medium [15], reducing the acquisition time by a factor equal to the number of transmitted beams. It has recently gained a lot of interest, especially for application in cardiac ultrasound imaging [15], where the possibility to observe and track heart motion in real-time is of the uttermost importance for diagnostic purposes. In [15], it has been shown that, by applying a Tukey apodization window on both transmit and receive sides, improved performance can be achieved thanks to the lowering of cross-talk artifacts that arise when multiple beams are transmitted simultaneously, but at the expense of lateral resolution. Other works in the literature have addressed this same problem, such as [16; 17; 18]. Our group has recently worked towards the joint use of MLT with a new non-linear beamforming algorithm [19], i.e., F-DMAS [8; 20; 21], and showed how F-DMAS overcomes the main problems that have so far limited the use of MLT in clinical practice (i.e., cross-talk artifacts and low lateral resolution). This algorithm basically consists of computing the aperture spatial autocorrelation in reception; however, the pulse-echo response is heavily jeopardized by MLT imaging [22], hence the question we address in this work is: does this affect spatial coherence? If so, how? To understand how spatial coherence varies in different imaging setups and with different image acquisition strategies can indeed be very useful, in that it allows us to explain the achievable performance and possible effects that could be observed in images generated with the previously mentioned spatial-coherence-based beamformers, which are obviously influenced by its behavior. In particular, here we are interested in analyzing the performance of F-DMAS when combined with different MLT strategies and when the spatial correlation operation, on which this algorithm is based, is only computed between signals within a certain maximum lag. Therefore, the work presented here is organized in two parts. In the first part, an analytical study of the spatial coherence trend in a simulated uniform medium with MLT imaging is presented.