An Application of Resistance Networks to the Problem of Adjustment by Least Squares

Abstract

IT is well known that the normal equations for the least squares adjustment of networks of spirit-levelling and similar observations can be solved very simply by relaxation¹ ; but the determination of the standard deviation of the n heights requires the solution of n sets of n simultaneous equations, and by any numerical method this is more laborious than the determination of the heights themselves. If σis the standard deviation of an observation of unit weight, the standard deviation of a height h_i is σω_i^1/2, where ω_i, the reciprocal of the weight of h_i, is the solution of an equation of the form A in this set of equations is the matrix of the coefficients of the normal equations and, therefore, known. Now, if a network of resistances is constructed which is topologically identical with the network of levelling, and is such that the admittance of each element is proportional to the weight of the corresponding observation, the admittance matrix will be proportional to A, and the above equation becomes the equation for the voltage appearing between the initial point and the ith point, when unit current is injected between these two points. Unlike Liebmann's applications of resistance networks to the solution of partial differential equations², no high accuracy is required because the weights of the observations are not sufficiently well known to justify it, and ordinary carbon resistors will usually be satisfactory.