Spatially explicit estimates of population density, together with appropriate estimates of uncertainty, are required in many management contexts. Density surface models (DSMs) are a two-stage approach for estimating spatially varying density from distance sampling data. First, detection probabilities—perhaps depending on covariates—are estimated based on details of individual encounters; next, local densities are estimated using a GAM, by fitting local encounter rates to location and/or spatially varying covariates while allowing for the estimated detectabilities. One criticism of DSMs has been that uncertainty from the two stages is not usually propagated correctly into the final variance estimates. We show how to reformulate a DSM so that the uncertainty in detection probability from the distance sampling stage (regardless of its complexity) is captured as an extra random effect in the GAM stage. In effect, we refit an approximation to the detection function model at the same time as fitting the spatial model. This allows straightforward computation of the overall variance via exactly the same software already needed to fit the GAM. A further extension allows for spatial variation in group size, which can be an important covariate for detectability as well as directly affecting abundance. We illustrate these models using point transect survey data of Island Scrub-Jays on Santa Cruz Island, CA, and harbour porpoise from the SCANS-II line transect survey of European waters. Supplementary materials accompanying this paper appear on-line.

在许多管理环境中，都需要对人口密度进行空间明确的估算，以及对不确定性的适当估算。密度表面模型（DSM）是一种从距离采样数据估计空间变化密度的两阶段方法。首先，检测概率（可能取决于协变量）是基于各个遭遇的详细信息进行估计的；接下来，使用GAM估算局部密度，方法是将局部遭遇率与位置和/或空间变化的协变量拟合，同时考虑估计的可检测性。对DSM的一种批评是，这两个阶段的不确定性通常不会正确地传播到最终方差估计中。我们展示了如何重新构造DSM，以便在GAM阶段将距离采样阶段的检测概率不确定性（无论其复杂程度）捕获为额外的随机效应。实际上，我们在拟合空间模型的同时对检测函数模型重新拟合了近似值。这样就可以通过完全适合GAM所需的相同软件直接计算总方差。进一步的扩展允许组大小的空间变化，这可能是可检测性的重要协变量，并直接影响丰度。我们使用加利福尼亚州圣克鲁斯岛的Island Scrub-Jays的点样点调查数据以及欧洲水域的SCANS-II线样点调查中的海豚来说明这些模型。本文随附的补充材料在线出现。