Estimation and imaging of linear and nonlinear propagation and scattering parameters in a material object where the material parameters for wave propagation and scattering has a nonlinear dependence on the wave field amplitude. The methods transmit at least two pulse complexes composed of co-propagating high frequency (HF) and low frequency (LF) pulses along at least one LF and HF transmit beam axis, where said HF pulse propagates close to the crest or trough of the LF pulse along at least one HF transmit beam, and where one of the amplitude and polarity of the LF pulse varies between at least two transmitted pulse complexes. At least one HF receive beam crosses the HF transmit beam at an angle, to provide at least two HF cross-beam receive signals from at least two transmitted pulse complexes with different LF pulses.