3-D Born–Fréchet traveltime kernel theory is recast in the context of scalar-wave propagation in a smooth acoustic medium, for simplicity. The predictions of the theory are in excellent agreement with ‘ground truth’traveltime shifts, measured by cross-correlation of heterogeneous-medium and homogeneous-medium synthetic seismograms, computed using a parallelized pseudospectral code. Scattering, wave-front healing and other finite-requency diffraction effects can give rise to cross-correlation traveltime shifts that are in significant disagreement with geometrical ray theory, whenever the cross-ath width of wave-peed heterogeneity is of the same order as the width of the banana—doughnut Fréchet kernel surrounding the ray. A concentrated off-ath slow or fast anomaly can give rise to a larger traveltime shift than one directly on the ray path, by virtue of the hollow-banana character of the kernel. The often intricate 3-D geometry of the sensitivity kernels of P, PP, PcP, PcP₂, PcP₃, … and P+pPwaves is explored, in a series of colourful cross-sections. The geometries of an absolute PPkernel and a differential PP— Pkernel are particularly complicated, because of the minimax nature of the surface-reflected PPwave. The kernel for an overlapping P+pPwave from a shallow-focus source has a banana–doughnut character, like that of an isolated P-wave kernel, even when the teleseismic pulse shape is significantly distorted by the depth phase interference. A numerically economical representation of the 3-D traveltime sensitivity, based upon the paraxial approximation, is in excellent agreement with the ‘exact’ ray-theoretical Fréchet kernel.