Ergodic sampling: Acquisition design to maximize information from limited samples

Data acquisition using equal spacing has been a standard practice in geophysics. The dense uniform sampling derived from Nyquist–Shannon sampling includes redundant samples, and it is sufficient but not necessary to adequately record target signals. We propose an ergodic sampling, which avoids the redundant samples in the dense uniform sampling and possesses the ability to capture the sufficiently similar information content as does Nyquist sampling. Ergodicity means that a key part of the system can represent the average performance of the entire system. Our ergodic samples are a critical subset of dense uniform samples and can represent the full uniform Nyquist samples. To find such a critical subset, we first examine the properties of different sampling patterns, including the sampling interval distribution, sampling angle distribution, areal sample density, and resolution in the spectral domain. We define the information sampling ability of sampling patterns based on these properties. The concept of information sampling ability that we have proposed serves as the criterion to compare different sampling patterns and assess their sampling performances. The sampling patterns with the same information sampling ability have the same capability to gather information, even though the appearance of sampling patterns may be different. We formulate an optimization problem to find this critical subset of sample locations, which has the fewest number of samples but has a similar information sampling ability as that of the desired dense uniform samples. This critical subset is irregularly located, has the optimized properties and forms the ergodic sampling pattern. We define this process of sampling design and associated understanding as the ergodic sampling. Ergodic sampling can be applied to gain two major benefits in practice. First, this approach can save a significant number of samples. We demonstrate ergodic sampling using one-dimensional synthetic data and a two-dimensional field geophysical dataset. The simulations confirm that, compared with other sampling strategies, ergodic sampling can use fewer samples to acquire the same amount of information, so that we can save cost. Alternatively, with the same budget, we can use the same number of samples through ergodic sampling to acquire more information. The new ergodic sampling can lead to a new generation of economic and efficient geophysical data acquisition, which could assist in increasing the discovery rate in resource exploration, tackling more earth science problems with a limited budget and can also benefit the environment in the process by reducing the invasiveness in potentially sensitive regions.