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This paper proposes an improvement on the scattering-integral (SI) approach for acoustic frequency-domain full waveform inversion (FWI) based on the individual Born kernels. The main development is a method for calculating the steepest-descent direction and the pseudo- Newton direction by vector operations with definite physical meaning, without needing to store the huge Fréchet kernels in memory beforehand. The Gauss-Newton descent direction can therefore be iteratively constructed without needing to store the huge approximate Hessian matrix or to calculate its inverse. The banded pseudo-Hessian and its inverse can be obtained in this way as well. This approach is efficient and makes the traditional SI approach more practical. At the same time, it keeps the advantages of the SI approach which can generate exact gradient of the objective function, enables separation of forward and inverse computation and provides straightforward access to Gauss-Newton iteration. This approach is called Born kernel full waveforminversion (BKFWI). Its effectiveness using the steepest-descent direction has been proved through 2-D numerical experiments. More fully resolved results and faster convergence are obtained when the Gauss-Newton direction is used. Because no additional forward simulations are needed for the Gauss-Newton direction, BKFWI is a highly valid alternative to traditional adjoint-state full waveform inversion (ADFWI) when the number of source stations is more than a few percent of the number of receiver stations. Such conditions are common in controlled source exploration, OBS (Ocean Bottom Seismometer) exploration, earthquake seismology, and even certain traditional seismic explorations. This method is also shown to be a convenient way to obtain accurate Gauss-Newton directions for multiparameter FWI because kernel coupling in the Hessian matrix can be easily handled. [ABSTRACT FROM AUTHOR]