Medical ultrasound imaging scanners typically display the envelope of the reflected signal on a log scale. The properties of this image and speckle patterns from collections of scatterers have a number of well-known disadvantages. One is the inability to differentiate between different scatterers that may have fundamentally different frequency-dependent scattering cross sections. This study proposes a framework for characterizing scattering behavior and visualizing the results as color coding of the B-scan image. The methodology matches a model of pulse-echo formation from typical situations to the mathematics of Gaussian weighted Hermite functions. The results show an ability to reveal some of the information otherwise hidden in the conventional envelope display, and can be generalized to more conventional bandlimited pulse functions. This new class of images is termed H-scan where 'H' stands for 'Hermite' or 'hue' to distinguish it from conventional B-scan format.