  Backus considered a boundary value problem for the Laplace equation with the
non-linear data in the form of the magnitude $|Du|$ of the gradient of the
solution u. We consider this problem with the data expanded by
$(\partial/\partial\nu)|Du|$ given on the boundary of the domain. To justify
the requirement for additional data, we use them to estimate the number of
sources for the related inverse source problem in the plane. We show that, for
an arbitrary dimension, a harmonic function satisfies a quasi-linear equation
on the boundary of the domain with the coefficients involving the augmented
data. We use the finite element method to recover the harmonic function on the
boundary by solving numerically the derived equation.
