Proposed approach:
Depending on the nature of the noise singular value decomposition (SVD) based approaches have been shown to be quite effective at signal extraction by identifying latent variables sources (i.e. variables that cannot or have not been directly measured).
Once noise has been removed one of a variety of regression based approaches would be suitable for attributing the left over web volume to TV commercial airings to determine web lifts.
Background:
My background is primarily in statistics, machine learning and data science and how these are applied to imaging, recommender systems and biological and physical sciences. My particular areas of interest have included the development of algorithms for pattern recognition in massive, multi-dimensional images (primarily hyperspectral and Raman), genomics, sparse coding, fitness tracker recommender systems and others.
Also, over the last three years I've gained extremely valuable experience as a co-founder and chief data scientist in a start-up. Our focus has been on the development of recommendation systems in the fitness space to help provide users insights into how they are progressing, what their relative strengths and weaknesses are and how they might address areas where additional work might be needed.