  We present an extensive grid of numerical simulations quantifying the
uncertainties in measurements of the Tip of the Red Giant Branch (TRGB). These
simulations incorporate a luminosity function composed of 2 magnitudes of red
giant branch (RGB) stars leading up to the tip, with asymptotic giant branch
(AGB) stars contributing exclusively to the luminosity function for at least a
magnitude above the RGB tip. We quantify the sensitivity of the TRGB detection
and measurement to three important error sources: (1) the sample size of stars
near the tip, (2) the photometric measurement uncertainties at the tip, and (3)
the degree of self-crowding of the RGB population. The self-crowding creates a
population of supra-TRGB stars due to the blending of one or more RGB stars
just below the tip. This last population is ultimately difficult, though still
possible, to disentangle from true AGB stars. In the analysis given here, the
precepts and general methodology as used in the Chicago-Carnegie Hubble Program
(CCHP) has been followed. However, in the Appendix, we introduce and test a set
of new tip detection kernels which internally incorporate self-consistent
smoothing. These are generalizations of the two-step model used by the CCHP
(smoothing followed by Sobel-filter tip detection), where the new kernels are
based on successive binomial-coefficient approximations to the
Derivative-of-a-Gaussian (DoG) edge detector, as is commonly used in modern
digital image processing.
