NURPS surfaces are the rational extension of Powell-Sabin splines in their normalized B-spline representation. In this paper we study the influence of modifying the weights of a NURPS surface. We describe the relation between the weights associated with a control triangle and the points on the NURPS surface by means of a double volume ratio. We also extend the concept of Farin points for rational Bézier curves to NURPS surfaces, resulting in weight points and weight triangles. They admit a local weight control in a geometrically intuitive way.