A systematic description of a new approach to determining the flow rates of liquids or gases transported through pipelines of circular cross sections is presented. The approach is based on multipath ultrasonic time-of-flight measurements. A mathematical technique for processing pulsed acoustic signals recorded in several measuring planes is developed. The technique provides precision estimates of flow rates owing to the use of quadrature integration methods. Several variants of realizing the multipath measurements are suggested. Tables of nodes and weights of the corresponding quadrature formulas, including both known values and values calculated for the first time, are presented.
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Tereshchenko et al. (2004) studied this question.