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Compressive sensing is a powerful method for reconstruction of sparsely-sampled data, based on statistical optimization. It can be applied to a range of flow measurement and visualization data, and in this work we show the usage in groundwater mapping. Due to scarcity of water in many regions of the world, including southwestern United States, monitoring and management of groundwater is of utmost importance. A complete mapping of groundwater is difficult since the monitored sites are far from one another, and thus the data sets are considered extremely “sparse”. To overcome this difficulty in complete mapping of groundwater, compressive sensing is an ideal tool, as it bypasses the classical Nyquist criterion. We show that compressive sensing can effectively be used for reconstructions of groundwater level maps, by validating against data. This approach can have an impact on geographical sensing and information, as effective monitoring and management are enabled without constructing numerous or expensive measurement sites for groundwater.

Fresh water is critical to sustainable living, economic growth, social stability and public health [

Compressive sensing is a powerful method that allows for the actual signals to be recovered from far fewer samples than what have been possible according to the Nyquist-Shannon sampling theorem [_{i} (e.g. Fourier series of sinusoidal functions).

f ( x ) = ∑ i = 1 n c i ψ i ( x ) (1)

For most signals, only a small number of the coefficients c_{i} are of significant magnitude, and most of the others can be discarded with negligible loss of information. Image compression works by having the full data, f(x), and evaluating the coefficients, c_{i}, a priori. Since the full image data are used during image compression, the entirety of f(x) is available for de-composition (Equation (1)) to find c_{i}, through Fourier transform, for example. From the set of coefficients, only the most significant c_{i} terms are retained, for later reconstruction, which is how image compression/reconstruction works. Compressive sensing takes advantage of this concept by attempting to find this coefficient set, c_{i}, in the absence of full data. Therefore, an algorithm is required to “recover” c_{i}, from under-sampled signal, f(x) in Equation (1), with the requirement that the coefficient matrix must have certain properties such as a small number of dominant terms. Due to under-sampled f(x), the equation or the system of equations such as Equation (1) is highly indeterminate with a constraint (a condition that the matrix must follow). For this type of problems, highly effective numerical methods such as l1-optimization, gradient pursuit or other convex optimization algorithms have been developed [

Compressive sensing is of course not without limits, and does need the following criteria to work, in applications such as groundwater mapping:

1) The desired groundwater network database is “compressible”, meaning that the discrete data (at sparsely-spaced probe locations) can be approximated as a series of orthogonal functions (e.g. sinusoidal function in a Fourier series). Based on working with most groundwater network data base, we find that this condition is satisfied since most continuous signals can be expressed reasonably accurately using truncated Fourier series.

2) The second condition relates to the coefficient matrix, c_{i} in Equation (1) as an example. This matrix needs to have a relatively small number of dominant terms, so that the indeterminate matrix can be numerically found under the “compressibility” constraint. Again, tests with groundwater database suggest that this condition is also satisfied. Technical details can be found in the Appendix and in the references cited above.

If the above criteria are met, then numerical algorithms such as the l1-optimization or total variational optimization routines can be used to find highly under-determined data matrix based on sparse input data. The step-by-step computational procedures used in this work follow Boyd and Vandenberghe [

For a demonstration of the compressive sensing technique, a sample MODFLOW image (256 by 256 pixel resolution or a total pixel count of 65,536) is used, for groundwater level in Fort Cobb Reservoir experimental watershed (780 km^{2}) [

The sampling density is progressively increased from top to bottom, in

We can track the rate at which the original data can be recovered at different sampling rates, and also test different data sampling geometries (pyramidal, radial and Cartesian), for the same MODFLOW data. The results are shown in

ErrorNorm = ∑ ( I i j − I i j T R U E ) 2 (2)

I_{ij}is the pixel value, relative to the actual data, I i j T R U E . We can easily see that the pyramidal sampling geometry far outperforms the others, and using this sampling method error norm of 10 or less is possible at only 10% sampling rate, meaning that 1/10^{th} of the data is necessary for a nearly full recovery of the data. Approximate reconstructions are possible at even less sampling rates. Thus, the ability to generate a full picture for the groundwater distribution is significantly enhanced using compressive sensing. This demonstration of the compressive sensing also points to the “compressibility” of groundwater data. As shown in

For the groundwater level measurements, we use the USGS database, augmented by data provided by the state water resources management in the southwestern states (California, Nevada, Arizona, and Utah). We first focus on the groundwater resources in the southwestern United States, where the recent historical drought conditions make the accurate assessments and management important. Compressive sensing can be applied in a similar manner to any other regions where minimal amount of measurement data are available. The groundwater level data are typically generated using a pressure transducer or a floating device in registered wells. In each registered well, depth-to-well (the water head) is measured [

For validation of the compressive sensing algorithm, we start with the data set as shown in

shown in

Thus,

the results with traditional methods of recovering missing data, e.g. Delaunay polygon, kriging, radial basis function, and inverse distance weighting methods. In hydrological monitoring, sparsity in the data is evidently a persistent problem, and several methods have been developed to “patch” this deficiency [

data. When the sampling rate is above 59%, as in

Delaunay polygon, kriging, radial basis function, and inverse distance weighting methods have been traditionally used to compensate for lack of data [

Now, we can check whether compressive sensing is able to construct two-dimensional mapping of the groundwater level. We start from the 0.25-degree resolution mapping in

as shown by the actual data in

We can see this effect when we increase the pixel resolution to 0.1˚ longitude × 0.1˚ latitude mapping by averaging the available data over smaller areas, as shown in

mapping obtained through compressive sensing shows many data cells that exhibit unreasonably low groundwater levels (blue cells). The method is able to only generate data near the points or cells at which there existed some amount of data.

We have shown that compressive sensing can be an effective method in hydrological monitoring, where the measurements are often sparsely populated. Important parameters, such as the groundwater level, across large regions can be mapped into a more complete contour data, at data sampling rates far below the Nyquist criterion. Although compressive sensing has been developed in image and signal processing, it evidently has significant applications in fluidic systems as well, such as groundwater hydrology since transport processes are continuous through physical principles of convection and diffusion. Fluidic processes mostly involve continuity and smoothness in the data (which is not necessarily the case in general imaging), which tend to satisfy the “compressibility” condition. Therefore, the potential for applications of compressive sensing in hydrology is quite wide and impactful.

This work was supported in part by the Ministry of Education (Czech Republic), under the program INTER_ EXCELLENCE (project number LTAUSA19053).

The authors declare no conflicts of interest regarding the publication of this paper.

Lee, T.-W., Lee, J.Y., Park, J.E., Bellerova, H. and Raudensky, M. (2021) Reconstructive Mapping from Sparsely-Sampled Groundwater Data Using Compressive Sensing. Journal of Geographic Information System, 13, 287-301. https://doi.org/10.4236/jgis.2021.133016

To illustrate the application of compressive sensing to groundwater data analysis, we can start with a simple data set composed of the size N by 1 matrix, and then N nodes are the elements of the groundwater level data. Each node acquires a sample x_{i} which is the averaged groundwater level at the specified coordinate ranges. Thus, we have a data set: x = [ x 1 , ⋯ , x n ] T . We can approximate data using a basis function (such as sinusoidal function in Fourier series), Ψ = [ ψ 1 , ⋯ , ψ n ] T . Then, the data can be found and it meets that x = ∑ i = 1 m z i ψ i as well as m ≪ n . Compressive sensing theory [

min ‖ x ‖ 1 such that y = Φ Ψ z

There are online, open-source programs for l1-optimization, such as Yang et al. [

Once one gets to this point, then extensions to two- or three-dimensional problems are straight-forward. For example, we can reconstruct a two-dimensional groundwater network data f ( t x , t y ) from samples F ( x , y ) of its discrete Fourier transform on a radial domain Θ. Two-dimensional Fourier transform is formulated below.

F ( x , y ) = ∑ t x = 0 N − 1 ∑ t y = 0 N − 1 f ( t x , t y ) e − 2 π i ( t x x N + t y y N )

where, f ( t x , t y ) is the two-dimensional groundwater level and F ( x , y ) represents sampled data. For two-dimensional data, l1-optimization algorithm is slow, and other methods such as total variation and gradient pursuit algorithms are used, also available as open-source codes [