Map construction:
The contours on these tropopause temperature maps are constructed by plotting the spatial location of the intersection of the 2.0 PVU surface with a particular isentrope. This method of construction is different from simply plotting the potential temperature as a single-valued function of q, because it allows folds in the tropopause to be seen as crossing contours.
Interpreting the maps:
Tropopause temperature maps are useful in interpreting tropospheric dynamics because of the association between boundary temperature anomalies and potential vorticity (PV) anomalies. In particular, quasi-geostrophic theory can be used to show that temperature anomalies on the upper and lower boundaries of a fluid can be thought of as collocated PV anomalies. On a lower boundary, the PV anomaly has the same sign as the temperature anomaly, while on an upper boundary the anomalies have opposite signs.
For example, a cold region of the tropopause is typically low in height and correlated with an underlying positive PV anomaly. A warm region at 850 mb (i.e. a region of anomalously high saturation equivalent potential temperature) is associated with an overlying positive PV anomaly. The distribution of tropopause and 850 mb temperatures can thus be thought of as distributions of PV on the upper and lower boundaries of the troposphere.
The vertical penetration depth of a particular boundary PV anomaly determines whether it can influence the properties of the other boundary. Extratropical cyclogenesis can be thought of as the interaction of upper- and lower-tropospheric PV anomalies (in the sense of the classical Eady model).
In the real atmosphere, PV is created and destroyed by diabatic processes. Some insight can be gained by examining the satellite images corresponding to a particular tropopause temperature map and noting how the tropopause temperatures evolve over time in both regions of active convection and regions of strong radiative cooling.