Mobile ground robots require perceiving and understanding their surrounding support surface to move around autonomously and safely. The support surface is commonly estimated based on exteroceptive depth measurements, e.g., from LiDARs. However, the measured depth fails to align with the true support surface in the presence of high grass or other penetrable vegetation. In this work, we present the Semantic Pointcloud Filter (SPF), a Convolutional Neural Network (CNN) that learns to adjust LiDAR measurements to align with the underlying support surface. The SPF is trained in a semi-self-supervised manner and takes as an input a LiDAR pointcloud and RGB image. The network predicts a binary segmentation mask that identifies the specific points requiring adjustment, along with estimating their corresponding depth values. To train the segmentation task, 300 distinct images are manually labeled into rigid and non-rigid terrain. The depth estimation task is trained in a self-supervised manner by utilizing the future footholds of the robot to estimate the support surface based on a Gaussian process. Our method can correctly adjust the support surface prior to interacting with the terrain and is extensively tested on the quadruped robot ANYmal. We show the qualitative benefits of SPF in natural environments for elevation mapping and traversability estimation compared to using raw sensor measurements and existing smoothing methods. Quantitative analysis is performed in various natural environments, and an improvement by 48% RMSE is achieved within a meadow terrain.