A new derivation of the temperature change estimation based on speckle-tracking estimation of axial echo shifts is presented which accounts for the nonuniform temperature baseline. The new method includes the spatial gradient of the temperature change in the thermal strain equation. In contrast to the previous method where thermal strain was calculated by accumulating the incremental displacement and differentiating along the axial direction, the solution to the new equation is shown to be found by using a differentiation-integration operator along the axial direction. Using a diagnostic probe and a High Intensity Focused Ultrasound (HIFU) transducer in Dual Mode Ultrasound Array (DMUA) setup the displacement data was collected from sub-therapeutic shots on the hind limb of a Copenhagen rat. The results of applying the new method showed a temperature change in a 2 mm axial extend which was consistent with the field scan pattern of the HIFU array.