A mathematical approach to improving the representation of surface water–groundwater exchange in the hyporheic zone

It is well known that land surface topography governs surface–groundwater interactions under some circumstances and can be separated in a Fourier-series spectrum that provides an exact analytical solution of both the surface and the underlying three-dimensional groundwater flows. We evaluate the performance of the current Fourier fitting process by testing on different scenarios of synthetic surfaces. We identify a technical gap and propose a new version of the approach which incorporates the spectral analysis method to help identify the statistically significant frequencies of the surface to guide the refinement and mesh. Our results show that spectral analysis is the method that can help improve the accuracy of representing the surface, thus further improving the accuracy of predicting the bedform-driven hyporheic exchange flows.


INTRODUCTION
Rivers are among the most fascinating freshwater bodies on Earth, integrating many forms of ecological processes that influence the water quality (Brunke &  However, the current stage of the limited knowledge of the connection between those individual river streambed features and the hyporheic zone cannot satisfy our need to optimize effective management of the multiscale river corridor system. Even though we have built many exciting and insightful hyporheic exchange models, we still have little predictive, transferable, confident understanding of the system so far (Ward & Packman ). This is because the controlling processes and mechanisms (Fehlman ; Elliott ; Elliott & Brooks a, b) used in all of these current models were based on simplified and empirical theories under certain assumptions and simplifications of natural bedform characteristics (Boano et al. ; Ward & Packman ). Even the most up-to-date model developer Susa acknowledged that the simplified representation of the streambed topography does not adequately characterize patterns and rates of hyporheic exchange (Stonedahl et al. ). Thus, we want to step further to investigate how bedform characteristics control the hyporheic exchange system in this proposed study.
In this study, the accuracy of Wörman et al.'s () analytical solution was tested on a very simple harmonic synthetic surface, the performance of the approach on var- By answering these research questions, here a novel version of the Fourier fitting process is developed. The novelty of this approach is incorporating the spectral analysis as a tool to guide us to fit the surface topography more accurately and reliably, and thus can further improve the prediction accuracy of hyporheic exchange metrics such as boundary fluxes in this study. We also demonstrate its ability to improve the accuracy of representing the surface and predicting the boundary fluxes, especially for those surfaces with characteristic scales and significant frequency signals.

METHOD
In this study, the research activities were carried out in three stages. We began by testing the performance of Wörman conductivity. Streamlines and residence time distributions can be calculated from the flow field using the numerical particle tracking technique.
In order to use the Fourier fitting to fit the surface topography, the amplitude coefficients are evaluated for a preassigned spectrum of k x ¼ 2π=λ x and k y ¼ 2π=λ y , where λ is wavelength (Wörman et al. ). Thus, the analytical solution can represent several topography observations in the matrix containing the pre-assigned harmonics of (1), H con- The synthetic surface used here is a simple harmonic synthetic surface with one frequency in both x-and ydirections shown in Figure 1 as where the amplitude of the surface h m is set as 30 (m), the wavelength of the surface λ is set the same for both the xand y-direction as λ ¼ 10 (m), thus the wave number for  they might have characteristic spatial scales. They showed the method of using spectral analysis to identify the characteristic spatial scales from the background signals (parallel to the example shown above is to identify the red points before assigning too many blue points to approximate the surface). Hence, we believe that the spectral analysis can be incorporated into the traditional Fourier fitting process to capture the characteristic bedform signals, further where k x and k y are the wave numbers in the x-and y-direction, and m and n are indices in the z array (x ¼ mΔx, y ¼ nΔy) (Perron et al. ). The element at (k x , k y ) in the DFT array corresponds to the two orthogonal frequency components: and the ranges of the wavenumbers are ÀN x k x (N x =2) À 1 and À(N y =2) À 1 k y (N y =2) (Perron et al. ). To collapse the 2D spectra into onedimensional (1D) plots, we adopt the radial frequency concept used in Perron et al.'s () work. The radial frequency is defined as: The power spectrum is estimated by the DFT periodogram which represents how the variance of z varies with frequency: We evaluate the significance of the signals of the topography data by assessing the P DFT of the DFT array. The Fourier transform also assumes that the input signal is periodic at the edges of the sampled interval ( Thus, for any 2D DFT applied by window function W(m, n), the equation becomes: jZ(k x, k y )j 2 (12)

Significance levels and significant signals identification
To use the power spectrum to evaluate the significance of these different signal components, we set up a confidence level where we can reject the null hypothesis that an observed periodic signal has occurred by chance in a random topographic surface, so that we can identify the sig- where χ 2 2 (1 À α) is the value at which the χ 2 cumulative distribution function with two degrees of freedom equals 1 À α. ). The 1D power spectra plot helps us to discriminate the differences between the surface signals from the generated background signals, thus answering the question of whether the surface has the characteristic spatial signals.
The 2D power spectra plot helps us to identify those characteristic signals of the surface. After obtaining the significant signals of the surface, we use an inverse Fourier transform process to reconstruct the characteristic spatial parts of the surface as well as the background noise part, thus decomposing the surface topography. Figure 3 shows the generic steps of conducting spectral analysis on a surface topography.
Fourier fitting process based on the results of the spectral analysis For any arbitrary surfaces, results obtained from the spectral analysis step are used to guide us in the following Fourier fitting process. Figure 4 shows the generic steps of how spectral analysis is incorporated into the original Fourier fitting process: if the surface is identified as a random surface that has no characteristic spatial scales, we directly apply the Fourier fitting process and assign the random frequencies to fit the surface; if the surface is identified as having characteristic spatial scales, we first retrieve and assign those characteristic signals of the surface to describe the surface, only apply the Fourier fitting process to fit the background (noise) part of the surface and combine two parts to generate final results.  It indicates that for the most part of the location, the mean error is less than 5% which is small. The mean error is calculated as:

RESULTS AND DISCUSSION
where A is the analytical solution value and N is the numeri-

CONCLUSIONS
In this study, a new Fourier fitting strategy that incorporates the spectral analysis as a tool to identify the characteristic scale of the surface and then applies the traditional Fourier fitting approach only on the random part of the surface was proposed. We also used a simple synthetic surface to verify that this new fitting strategy will not only improve the sur- (2) The accuracy of approximating the synthetic surfaces using the traditional Fourier fitting process does not always increase with the increasing number of incorporated Fourier fitting frequencies. Instead, the influence of the number of incorporated Fourier fitting frequencies on approximating surface varies and depends on the shape and complexity of the surfaces to be approximated.
(3) It is the absence of the bedform characteristic signals in the current Fourier fitting process that leads to the absence of the characteristic hyporheic exchange patterns in the current multiscale model system. The absence of the bedform characteristic signals can be retrieved and identified by spectral analysis.
(4) Our new proposed approach is effective in improving the accuracy of representing the surface and predicting the hyporheic exchange flux. The new proposed method will be more efficient when dealing with regulated surface topography than random surface topography.