The κ parameter (Anderson and Hough, 1984), and namely its site‐specific component (κ0), is important for predicting and simulating high‐frequency ground motion. We develop a framework for estimating κ0 and addressing uncertainties under the challenging conditions often imposed in practice: (1) low seismicity (limited, poor‐quality, distant records); (2) limited‐bandwidth data from the Transportable Array (TA; maximum usable frequency 16 Hz); and (3) low magnitudes (ML 1.2–3.4) and large uncertainty in stress drop (corner frequency). We cannot resolve stress drop within the bandwidth, so we propose an approach that only requires upper and lower bounds on its regional values to estimate κ0. To address uncertainties, we combine three measurement approaches (acceleration spectrum slope [AS]; displacement spectrum slope [DS]; and broadband spectral fit). We also examine the effect of crustal amplification, and find that neglecting it can affect κ0 by up to 35%. DS estimates greatly exceed AS estimates. We propose a reason behind this bias, related to the residual effect of the corner frequency on κAS and κDS. For our region, we estimate a frequency‐independent mean S‐wave Q of 900±300 at 9–16 Hz, and an ensemble mean κ0 over all sites of 0.033±0.014 s. This value is similar to the native κ0 of the Next Generation Attenuation‐West 2 ground‐motion prediction equations, indicating that these do not need to be adjusted for κ0 for use in southern Arizona. We find that stress‐drop values in this region may be higher compared to estimates of previous studies, possibly due to trade‐offs between stress drop and κ0. For this dataset, the within‐approach uncertainty is much larger than the between‐approach uncertainty, and it cannot be reduced if the data quality is not improved. The challenges discussed here will be relevant in studies of κ for other regions with bandlimited data; for example, any region where data come primarily from the TA.