  The Davenport spectrum is a modification of the classical Kolmogorov spectrum
for the inertial range of turbulence that accounts for non-scaling low
frequency behavior. Like the classical fractional Brownian motion vis-\`a-vis
the Kolmogorov spectrum, tempered fractional Brownian motion (tfBm) is a
canonical model that displays the Davenport spectrum. The autocorrelation of
the increments of tfBm displays semi-long range dependence (hyperbolic and
quasi-exponential decays over moderate and large scales, respectively), a
phenomenon that has been observed in wide a range of applications from wind
speeds to geophysics to finance. In this paper, we use wavelets to construct
the first estimation method for tfBm and a simple and computationally efficient
test for fBm vs tfBm alternatives. The properties of the wavelet estimator and
test are mathematically and computationally established. An application of the
methodology to the analysis of geophysical flow data shows that tfBm provides a
much closer fit than fBm.
