### Bridge between the two estimates

The pointing-list method also gives two useful diagnostics that bridge it to the area-based estimate. First, the imaged area is computed directly from the non-blanked pixels:

| (69)

In dual-band mode, and are reported independently. If , the primary beam is larger at than at . For the same pointing offsets and the same primary-beam truncation level, the valid area at the highest frequency is therefore included in the valid area at the lowest frequency. The common overlap is thus simply the highest-frequency imaged area,

| (70)

The overlap can then be reported as percentages of each band area:

| (71)

The common-overlap quantities are not used to define the median sensitivity, which remains a per-band quantity computed on each band's own valid area.

A purely geometrical number of independent primary beams can then be formed from this imaged area:

| (72)

This quantity is useful for comparing the pointing-list footprint to the area originally requested by the user, but it is not, by itself, a sensitivity-equivalent number of beams.

The sensitivity-equivalent bridge is obtained by asking which value of would make the area-based formula reproduce the median sensitivity of the actual pointing pattern at frequency , for the same observing setup. Equating the expressions and yields

| (73)

The quantity () is the number of independent primary beams that would make the area-based TSE formula give the same median sensitivity as the actual pointing list at frequency . It is different from : the former is defined by median sensitivity, whereas the latter is defined only by the imaged area.

In summary, the pointing-list estimate can also be understood as a minimal modification of the nine-step summary of the area-based estimate above. The observing overhead part of the computation is intentionally kept unchanged. In particular, Steps 3 to 8 keep the same equations once the map area and the number of pointings have been defined. The PMS-specific changes are confined to a new step 0 plus modified steps 1, 2, and 9.

0.
  Before applying the observing-time recipe, the actual pointing offsets are used to compute the dimensionless mosaic noise factor at the reference frequency of each receiver band

  | (74)

  Here is the primary beam of pointing , normalized to unity at its center and evaluated at sky position . Pixels outside the primary-beam support defined by the adopted truncation level are blanked. This defines the valid imaged area

  | (75)

  and the typical noise factor

  | (76)

  where is the set of non-blanked pixels at frequency . The minimum and maximum values of over the same area are also reported to describe the sensitivity variation between the best-covered part of the mosaic and its edges.
1.
  The user-provided area used by the TSE estimate is replaced by the imaged area measured from the pointing-list sensitivity map

  | (77)

  This area is frequency dependent because the primary beam and thus the non-blanked support depend on the reference frequency of the receiver band. It is this area that is used whenever the PMS pointing-list estimate needs a map area, for instance in the scan-time condition.
2.
  The number of pointings is no longer inferred from the standard hexagonal sampling relation. It is the actual number of offsets in the PMS pointing list

  | (78)

  The relation

  | (79)

  is therefore only used by the quick area-based estimate, not by the pointing-list estimate.
9.
  The rms is first computed for one pointing of the actual mosaic

  | (80)

  and

  | (81)

  The typical mosaic sensitivity is then obtained by multiplying this per-pointing rms by the median noise factor of the actual pointing pattern

  | (82)

  Equivalently, for comparison with the area-based formula, this last step can be written by replacing in Eq. by the sensitivity-equivalent number of beams

  | (83)

  This equivalent form is only a bridge to the TSE formulation; the pointing-list computation itself keeps the explicit separation between the per-pointing rms and the mosaic noise factor.