Notice that in the diagram, three of the four vertices on the "river bottom" at A are connected to the same vertex at the left side of the "bottom" of B. If you were to connect the vertices differently, for example connect three of the vertices of A to the "right bottom" vertex of B, you would form a solid with a larger volume.
Given that you do not really know how the actual cross-section of the river varies between the measured cross-sections, you have to take a guess. You _could_ linearly interpolate the cross-sectional area, but if you apply that method to several cross-sections parallel to the base of a pyramid, you don't get back the volume of the pyramid. An alternative might be to use the formula for the volume of a frustum:
$$ V = \frac13 h(A + B + \sqrt{AB})$$
where $A$ and $B$ are the areas of the two bases of the frustum (corresponding to the two cross-sections A and B in your figure) and $h$ is the distance between cross-sections.