A method for estimating locally varying nonlinear elastic parameters, NEPs, in a spatial estimation region, SER, of a material, from a set of measured nonlinear propagation delays, NPDs, as a function of position in said SER of the material, utilizing a mathematical model that from spatially varying NEP values in said SER produces spatially varying model NPDs in said SER, said method comprises a) measuring the NPDs as a function of position in said SER through - transmitting at least two pulse complexes composed of overlapping low frequency, LF, and a high frequency, HF, pulses, along LF and HF transmit beams that overlap at least within said SER, and - recording HF receive signals from scattered HF pulses from object structures within a set of depth regions along HF transmit beams within said SER from at least two transmitted pulse complexes with different LF pulses along the same HF transmit beam, and - comparing HF receive signals scattered from said HF pulses of pulse complexes with different LF pulses, from said set of depth regions along the same HF transmit beam, to produce estimates of said NPDs at said set of depth regions along said HF transmit beams in said SER, and b) forming an estimation functional, EF, that with a set of NEPs as input gives a weighted sum of i) a distance function of the difference between said measured NPDs and said model NPDs for said NEP values, weighted by a spatially variable distance weight, and ii) a measure of the local variation of said NEP values within said SERs weighted by a spatially variable variation weight, and where c) local values of said distance weight and said variation weight are estimated from one or both of i) an assessment of strong local reflectors in the received HF signals, and ii) an assessment of the local magnitude of multiple scattered HF noise relative to the local magnitude of the first order scattered HF signal, and d) for a given set of measured NPDs, minimizing said EF with respect to said set of NEPs, and using the set of NEPs that minimizes said EF to form an estimate of spatially varying NEPs within said SER. a) measuring the NPDs as a function of position in said SER through - transmitting at least two pulse complexes composed of overlapping low frequency, LF, and a high frequency, HF, pulses, along LF and HF transmit beams that overlap at least within said SER, and - recording HF receive signals from scattered HF pulses from object structures within a set of depth regions along HF transmit beams within said SER from at least two transmitted pulse complexes with different LF pulses along the same HF transmit beam, and - comparing HF receive signals scattered from said HF pulses of pulse complexes with different LF pulses, from said set of depth regions along the same HF transmit beam, to produce estimates of said NPDs at said set of depth regions along said HF transmit beams in said SER, and - transmitting at least two pulse complexes composed of overlapping low frequency, LF, and a high frequency, HF, pulses, along LF and HF transmit beams that overlap at least within said SER, and - recording HF receive signals from scattered HF pulses from object structures within a set of depth regions along HF transmit beams within said SER from at least two transmitted pulse complexes with different LF pulses along the same HF transmit beam, and - comparing HF receive signals scattered from said HF pulses of pulse complexes with different LF pulses, from said set of depth regions along the same HF transmit beam, to produce estimates of said NPDs at said set of depth regions along said HF transmit beams in said SER, and b) forming an estimation functional, EF, that with a set of NEPs as input gives a weighted sum of i) a distance function of the difference between said measured NPDs and said model NPDs for said NEP values, weighted by a spatially variable distance weight, and ii) a measure of the local variation of said NEP values within said SERs weighted by a spatially variable variation weight, and where i) a distance function of the difference between said measured NPDs and said model NPDs for said NEP values, weighted by a spatially variable distance weight, and ii) a measure of the local variation of said NEP values within said SERs weighted by a spatially variable variation weight, and where c) local values of said distance weight and said variation weight are estimated from one or both of i) an assessment of strong local reflectors in the received HF signals, and ii) an assessment of the local magnitude of multiple scattered HF noise relative to the local magnitude of the first order scattered HF signal, and d) for a given set of measured NPDs, minimizing said EF with respect to said set of NEPs, and using the set of NEPs that minimizes said EF to form an estimate of spatially varying NEPs within said SER.