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We analyze the basic techniques for the investigation of the deep structure of the mantle and the shortcomings of the models of mantle structures derived from them. Thus, we reveal that there is no analysis of the velocity field by means of analytical transformants. Therefore, we developed and tested a new approach to define the mantle boundaries based on the calculations of the sequence of P-waves velocity derivatives. As a result, we obtain some new set of velocity gradient distributions for the principal tectonic structures of the Ukrainian Shield along the composite profile. The boundaries of the mantle discontinuities according to the velocity gradient we define in a special manner to eliminate the false anomalies and the fluctuations of the velocity curves that occur due to the conversion of the hodograph into the mean velocities. The smoothing of the velocity curve we perform with a previously defined wavelength step being equal to 50 km. We treat the calculated velocity gradient anomalies as the useful signal response above the appropriate sections, which have different velocity accelerations levels inside the upper mantle. We assume that the mantle anomalies have the same physical background (density/viscosity distributions, temperature gradients etc.) within each range with the equal acceleration value. However, the singular points determined by the inflections of the gradient curve could be the possible boundaries of additional inhomogeneities within the mantle. We calculate both the 1st and the 2nd derivatives for the velocity curves obtained. The excesses 2.5-D model of the 1-th and 2-th gradient curves (the acceleration of the gradvp itself) determine the position of the max / min anomalies of gradvp at the consolidated seismic profile within the Ukrainian Shield. Finally, we analyze in detail the distribution of velocity gradients of P-waves within the upper mantle in the depth range of 50–750 km. It results in the identification of a series of additional gradient velocity boundaries within three principal structural horizons of the upper mantle (under ~ 200–300 km, ~ 410–500 km, and ~ 600–650 km respectively).

2.5 dimensional model of mantle heterogeneities under the Ukrainian shield according to the gradients of the velocities of seismic waves