Volterra series was applied to river runoff phenomena to model the input-output nonlinear relation between rainfall and runoff of rivers. The application results were reported in “Electrical Engineering in Japan” (vol. 98, 101). In the report, good results from modeling and simulation were shown, meanwhile unfavorable results were also shown in some cases depending on modeling condition.

The cause of unfavorable results was inferred as due to ill-conditioning of the coefficient matrix in the linear equation which was derived by expanding the Volterra kernels with the Laguerre function system. To solve this, advanced research was then conducted and reported in another publication where rank-reduced singular value decomposition was applied and adequate results were obtained.

Thus, in this paper, the causes of ill-conditioning have been examined in the cases of impulse input and usual input to the 2^nd order Volterra model. As a result, the fact that the coefficient matrix is of ill-conditioning has been verified mathematically.