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We observed with ALMA three deeply buried nuclei in two galaxies, NGC 4418and Arp 220, at $\sim$0.2$''$ resolution over a total bandwidth of 67 GHz in$f_{\rm rest}$ = 215 - 697 GHz. Here we (1) introduce our program, (2) describeour data reduction method for wide-band, high-resolution imaging spectroscopy,(3) analyze in visibilities the compact nuclei with line forests, (4) develop acontinuum-based estimation method of dust opacity and gas column density inheavily obscured nuclei, which uses the BGN (buried galactic nuclei) model andis sensitive to $\log(N_{\rm H_2}/{\rm cm}^{-2}) \sim $ 25 - 26 at $\lambda\sim 1$ mm, and (5) present the continuum data and diagnosis of our targets.The three continuum nuclei have major-axis FWHM of $\sim$0.1$''$-0.3$''$(20-140 pc) aligned to their rotating nuclear disks of molecular gas. However,each nucleus is described better with two or three concentric components thanwith a single Gaussian. The innermost cores have sizes of 0.05$''$-0.10$''$(8-40 pc), peak brightness temperatures of ~100-500 K at 350 GHz, and morefractional flux at lower frequencies. The intermediate components correspond tothe nuclear disks. They have axial ratios of $\approx$0.5 and henceinclinations $\stackrel{>}{\sim} 60$ deg. The outermost elements include thebipolar outflow from Arp 220W. We estimate 1 mm dust opacity of $\tau_{\rmd,1mm} \approx 2.2$, $1.2$, and $\stackrel{<}{\sim} 0.4$ respectively for NGC4418, Arp 220W, and Arp 220E. The first two correspond to $\log(N_{\rm H}/{\rmcm}^{-2}) \sim 26$ for conventional dust-opacity laws, and hence the nuclei arehighly Compton thick.

Deeply Buried Nuclei in the Infrared-Luminous Galaxies NGC 4418 and Arp 220: I. ALMA Observations at $\lambda = $1.4-0.4 mm and Continuum Analysis