  The estimation of commuting flows at different spatial scales is a
fundamental problem for different areas of study. Many current methods rely on
parameters requiring calibration from empirical trip volumes. Their values are
often not generalizable to cases without calibration data. To solve this
problem we develop a statistical expression to calculate commuting trips with a
quantitative functional form to estimate the model parameter when empirical
trip data is not available. We calculate commuting trip volumes at scales from
within a city to an entire country, introducing a scaling parameter alpha to
the recently proposed parameter free radiation model. The model requires only
widely available population and facility density distributions. The parameter
can be interpreted as the influence of the region scale and the degree of
heterogeneity in the facility distribution. We explore in detail the scaling
limitations of this problem, namely under which conditions the proposed model
can be applied without trip data for calibration. On the other hand, when
empirical trip data is available, we show that the proposed model's estimation
accuracy is as good as other existing models. We validated the model in
different regions in the U.S., then successfully applied it in three different
countries.
