= Solution
For the <two-zone cloudy annulus transmission model>, assume a uniform stellar disc, an opaque planet below $R_p$, $0\leq H_c\leq H$, and constant total slant <optical depth> within each radial zone. Neglect scattered light entering the beam and the planet's own emission. A ray transmits the fraction $e^{-\tau}$, so each zone blocks its projected area multiplied by $1-e^{-\tau}$.
The opaque disc, cloudy annulus and clear annulus have areas $\pi R_p^2$, $\pi[(R_p+H_c)^2-R_p^2]$ and $\pi[(R_p+H)^2-(R_p+H_c)^2]$, respectively. Thus the <exoplanet transmission spectrum>, expressed as total transit depth, is
$$
\boxed{\begin{aligned}
\delta_\lambda=1-\frac{F_{\rm in,\lambda}}{F_{\rm out,\lambda}}
={}&\frac{R_p^2}{R_s^2}
+\frac{(R_p+H_c)^2-R_p^2}{R_s^2}(1-e^{-\tau_{\lambda,c}})\\
&+\frac{(R_p+H)^2-(R_p+H_c)^2}{R_s^2}(1-e^{-\tau_\lambda}).
\end{aligned}}
$$
Here $\tau_{\lambda,c}$ means the total <optical depth> on rays assigned to the cloudy zone. If it denotes cloud extinction alone, its exponent must instead contain the sum of cloud and gas <optical depths>. For a geometrically thin <exoplanet atmosphere>, the two atmospheric prefactors become $2R_pH_c/R_s^2$ and $2R_p(H-H_c)/R_s^2$.
An opaque <exoplanet cloud deck> gives the simpler expression
$$
\delta_\lambda=\frac{(R_p+H_c)^2}{R_s^2}+\frac{(R_p+H)^2-(R_p+H_c)^2}{R_s^2}(1-e^{-\tau_\lambda}).
$$
The <exoplanet cloud deck> raises the wavelength-independent occulting radius and suppresses the clear atmospheric contribution. If $H_c=H$, the spectrum is flat at $(R_p+H)^2/R_s^2$; if $H_c=0$, the ordinary <annulus model for transmission spectroscopy> is recovered. Stellar limb darkening and varying tangent-ray <optical depth> would require an intensity-weighted radial integral rather than this constant-depth model.
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