Recent studies have shown how the distribution of $^{56}$Ni within the ejecta of type Ia supernovae can have profound consequences on the observed light curves. Observations at early times can therefore provide important details on the explosion physics in thermonuclear supernovae. We present a series of radiative transfer calculations that explore variations in the $^{56}$Ni distribution. Our models also show the importance of the density profile in shaping the light curve, which is often neglected in the literature. Using our model set, we investigate the observations that are necessary to determine the $^{56}$Ni distribution as robustly as possible within the current model set. We find that this includes observations beginning at least $sim$14 days before $B$-band maximum, extending to approximately maximum light with a high ($lesssim$3 day) cadence, and in at least one blue and one red band are required (such as $B$ and $R$, or $g$ and $r$). We compare a number of well-observed type Ia supernovae that meet these criteria to our models and find that the light curves of $sim$70-80% of objects in our sample are consistent with being produced solely by variations in the $^{56}$Ni distributions. The remaining supernovae show an excess of flux at early times, indicating missing physics that is not accounted for within our model set, such as an interaction or the presence of short-lived radioactive isotopes. Comparing our model light curves and spectra to observations and delayed detonation models demonstrates that while a somewhat extended $^{56}$Ni distribution is necessary to reproduce the observed light curve shape, this does not negatively affect the spectra at maximum light. Investigating current explosion models shows that observations typically require a shallower decrease in the $^{56}$Ni mass towards the outer ejecta than is produced for models of a given $^{56}$Ni mass.
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