We reconstruct the elastic modulus distribution for one tissue mimicking (TM) phantom and two in vivo biopsy-confirmed liver tumors using curvilinear ultrasound echo data. Spatial distribution of the relative elastic modulus values is determined by solving an inverse problem within a region of interest (ROI). This inverse problem solution requires knowledge of the ultrasonically measured displacement field in a uniform rectilinear grid to ensure that the resolution on the resultant relative elastic modulus elastogram will be uniform over the entire ROI. Taking advantage of a new speckle tracking algorithm, two different displacement tracking strategies are investigated: 1) sector-shaped ultrasound data were converted to ultrasound data on a rectilinear grid prior to speckle tracking and 2) axial and lateral displacements directly obtained from sector-shaped data were converted to vertical and horizontal displacements on a rectilinear grid after speckle tracking. Compared with strain elastography (SE), TM phantom results show that relative elastic modulus imaging (REMI) using Strategy 2 provided higher contrast-to-noise ratios (>300% and 25% increases compared with SE and REMI using Strategy 1, respectively). Furthermore, in phantoms, REMI using Strategy 2 more accurately (a 1.3% difference to shear wave elastography measurements) estimated the elastic contrast ratio between the target and the background, compared with both SE (20%-25%) and REMI using Strategy 1 (4.1%). It was also observed that relative modulus elastograms were more consistent with anatomical structures visualized on corresponding B-mode images for the two in vivo liver cases. Overall, we conclude that applying REMI is feasible for abdominal organs such as the liver. Strategy 2 offered improved and consistent results for the data investigated.