No abstract

Inversion of geophysical data relies on knowledge about how to solve the forward problem, that is, computing data from a given set of model parameters. In many applications of inverse problems, the solution to the forward problem is assumed to be known perfectly, without any error. In reality, solving the forward model (forward-modeling process) will almost always be prone to errors, which we referred to as modeling errors. For a specific forward problem, computation of crosshole tomographic first-arrival traveltimes, we evaluated how the modeling error, given several different approximate forward models, can be more than an order of magnitude larger than the measurement uncertainty. We also found that the modeling error is strongly linked to the spatial variability of the assumed velocity field, i.e., the a priori velocity model. We discovered some general tools by which the modeling error can be quantified and cast into a consistent formulation as an additive Gaussian observation error. We tested a method for generating a sample of the modeling error due to using a simple and approximate forward model, as opposed to a more complex and correct forward model. Then, a probabilistic model of the modeling error was inferred in the form of a correlated Gaussian probability distribution. The key to the method was the ability to generate many realizations from a statistical description of the source of the modeling error, which in this case is the a priori model. The methodology was tested for two synthetic ground-penetrating radar crosshole tomographic inverse problems. Ignoring the modeling error can lead to severe artifacts, which erroneously appear to be well resolved in the solution of the inverse problem. Accounting for the modeling error leads to a solution of the inverse problem consistent with the actual model. Further, using an approximate forward modeling may lead to a dramatic decrease in the computational demands for solving inverse problems.

No abstract

Least‐squares tomographic inversion of cross‐borehole ground penetrating radar (GPR) data was used to estimate the radar wave velocity distribution in the upper ∼10 m of the vadose zone at a field site in northern Zealand, Denmark. The radar wave velocities were transformed to values of water saturation and formed a basis for hydrologic studies of flow characteristics of the vadose zone. Cross‐borehole GPR data are likely to be contaminated by correlated data errors that may give rise to significant artifacts in the inverse estimate of the velocity distribution if they are not properly accounted for during the inversion process. We analyzed two sources of correlated data errors (unknown cavities and small‐scale clay‐enriched zones close to the borehole walls), which we assumed to play a significant role in the cross‐borehole GPR data sets collected at our field site. The correlated errors may be accounted for by specification of data error covariance matrices, which are included as a priori knowledge in the inverse operator used in the tomographic algorithm. The study indicates that proper accounting for correlated data errors significantly suppresses the effects of these data errors and results in trustworthy inverse estimates of the radar wave velocities between the boreholes. Suppression of the influence of the correlated data errors is a necessity for obtaining realistic hydrologic models of the vadose zone. Using popular inverse methods that disregard the data error correlation properties can lead to severe artifacts in the inverse model estimate and, therefore, can lead to incorrect hydrologic interpretations. Our findings are based on both synthetic tests and a real data example.