  The Helioseismic Magnetic Imager (HMI) onboard the Solar Dynamics Observatory
(SDO) records line-of-sight Dopplergram images of convective flows on the
surface. These images are used to obtain the multi-scale convective spectrum.
We design a pipeline to process the raw images to remove large-scale features
like differential rotation, meridional circulation, limb shift and imaging
artefacts. The Hierarchical Equal Area Pixelization scheme (HEALPix) is used to
perform spherical harmonic transforms on the cleaned image. Because we only
have access to line-of-sight velocities on half the solar surface, we define a
"mixing matrix" to relate the observed and true spectra. This enables the
inference of poloidal and toroidal flow spectra in a single step through the
inversion of the mixing matrix. Performing inversions on a number of flow
profiles, we find that the poloidal flow recovery is most reliable among all
the components. We also find that the poloidal spectrum is in qualitative
agreement with inferences from Local Correlation Tracking (LCT) of granules.
The fraction of power in vertical motions increases as a function of wavenumber
and is at 8% level for $\ell = 1500$. In contrast to seismic results and LCT,
the flows show nearly no temporal-frequency dependence. Poloidal flow power
peaks in the range of $\ell - |m| \approx 150-250$ which may potentially hint
at a latitudinal preference for convective flows.
