  We investigate the connection between a cluster's structural configuration
and observable measures of its gas emission that can be obtained in X-ray and
Sunyaev-Zeldovich (SZ) surveys. We present an analytic model for the
intracluster gas density profile: parameterised by the dark matter halo's inner
logarithmic density slope, $\alpha$, the concentration, $c$, the gas profile's
inner logarithmic density slope, $\varepsilon$, the dilution, $d$, and the gas
fraction, $\eta$, normalised to cosmological content. We predict four probes of
the gas emission: the emission-weighted, $T_\mathrm{X}$, and mean gas
mass-weighted, $T_\mathrm{m_g}$, temperatures, and the spherically,
$Y_\mathrm{sph}$, and cylindrically, $Y_\mathrm{cyl}$, integrated Compton
parameters. Over a parameter space of clusters, we constrain the X-ray
temperature scaling relations, $M_{200} - T_\mathrm{X}$ and $M_{500} -
T_\mathrm{X}$, within $57.3\%$ and $41.6\%$, and $M_{200} - T_\mathrm{m_g}$ and
$M_{500} - T_\mathrm{m_g}$, within $25.7\%$ and $7.0\%$, all respectively. When
excising the cluster's core, the $M_{200} - T_\mathrm{X}$ and $M_{500} -
T_\mathrm{X}$ relations are further constrained, to within $31.3\%$ and
$17.1\%$, respectively. Similarly, we constrain the SZ scaling relations,
$M_{200} - Y_\mathrm{sph}$ and $M_{500} - Y_\mathrm{sph}$, within $31.1\%$ and
$17.7\%$, and $M_{200} - Y_\mathrm{cyl}$ and $M_{500} - Y_\mathrm{cyl}$, within
$25.2\%$ and $22.0\%$, all respectively. The temperature observable
$T_\mathrm{m_g}$ places the strongest constraint on the halo mass, whilst
$T_\mathrm{X}$ is more sensitive to the parameter space. The SZ constraints are
sensitive to the gas fraction, whilst insensitive to the form of the gas
profile itself. In all cases, the halo mass is recovered with an uncertainty
that suggests the cluster's structural profiles only contribute a minor
uncertainty in its scaling relations.
