In this paper, a two-dimensional (2D) quantitative sonoelastographic technique for estimating local shear wave speeds from slowly propagating shear wave interference patterns (termed crawling waves) is presented. Homogeneous tissue-mimicking phantom results demonstrate the ability of quantitative sonoelastographic imaging to accurately reconstruct the true underlying shear wave speed distribution as verified using mechanical measurements. From heterogeneous phantoms containing a 5 or 10 mm stiff inclusion, results indicate that increasing the estimator kernel size increases the transition zone length about boundaries. Contrast-to-noise ratio (CNR) values from quantitative sonoelastograms obtained in heterogeneous phantoms reveal that the 2D quantitative sonoelastographic imaging technique outperforms the one-dimensional (ID) precursor in terms of image noise minimization and contrast enhancement. Experimental results from an embedded porcine liver specimen with an induced radiofrequency ablation (RFA) lesion validate 2D quantitative sonoelastographic imaging in tissue. Overall, 2D quantitative sonoelastography was shown to be a promising new imaging method to characterizing the shear wave speed distribution in elastic materials.