Technical Note: Partial wavelet coherency for improved understanding of 1 scale-specific and localized bivariate relationships in geosciences 2

8 Bivariate wavelet coherency is widely used to untangle the scale-specific and localized 9 bivariate relationships in geosciences. However, it is well-known that bivariate 10 relationships can be misleading when both variables are correlated to other variables. Partial 11 wavelet coherency (PWC) has been proposed, but is limited to one excluding variable and 12 presents no phase information. We aim to develop a new PWC method that can deal with 13 multiple excluding variables and presents phase information for the PWC. Tests with both 14 stationary and non-stationary artificial datasets verified the known scale- and localized 15 bivariate relationships after eliminating the effects of other variables. Compared with the 16 previous PWC method, the new method has the advantages of capturing phase information, 17 dealing with multiple excluding variables, and producing more accurate results. The new 18 method was also applied to two field measured datasets. Results showed that the coherency 19 2 between response and predictor variables was usually less affected by excluding variables 20 when predictor variables had higher correlation with the response variable. Application of 21 the new method also confirmed the best predictor variables for explaining temporal 22 variations in free water evaporation at Changwu site in China and spatial variations in soil 23 water content in a hummocky landscape in Saskatchewan Canada. We suggest the PWC 24 method to be used in combination with previous wavelet methods to untangle the scale- 25 specific and localized multivariate relationships in geosciences. The PWC calculations 26 were coded with Matlab and are available in the supplement.

In the case of one excluding variable, the numerators between Eqs. (9) and (14) differ, 121 but the denominators remain the same.  124 The PWC is first tested using the cosine-like artificial dataset produced following Yan 125 and Gao (2007). The cosine-like artificial datasets are suitable for testing the new method 126 because they mimic many spatial or temporal series in geoscience such as climatic variables, 127 hydrologic fluxes, seismic signals, El Niño-Southern Oscillation, land surface topography, 128 ocean waves, and soil moisture. The procedures to test the PWC is largely based on Hu and 129 https://doi.org/10.5194/hess-2020-315 Preprint. Discussion started: 7 August 2020 c Author(s) 2020. CC BY 4.0 License. Supplement) are selected from the five cosine waves (e.g., y1 to y5 or z1 to z5) or their 140 derivatives. The exact variables and procedures to test the new PWC method are explained 141 later on. 142 The PWC between response variable y (or z) and predictor variable, i.e., y2 (or z2), is first 1 (moderate noise), and 4 (high noise) are added to y2 (or z2) as suggested by Hu and Si 149 (2016) to simulate non-perfect cyclic patterns of the excluding variables. They are referred 150 to as y2wn (or z2wn), y2mn (or z2mn), and y2sn (or z2sn), respectively; and (d) a combination of type b and type c. They are referred to as y2wnh0 (or z2wnh0), y2mnh0 (or z2mnh0), and 152 y2snh0 (or z2snh0), respectively. The same data are also analyzed using the Mihanović et al.

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The PWC between response variable y (or z) and predictor variable, i.e., y2y4 (sum of y2 155 and y4) for the stationary case or z2z4 (sum of z2 and z4) for the non-stationary case, is 156 calculated with two excluding variables, which is a combination of y4 (or z4) and y2 (or z2) 157 or its noised series (y2wn or z2wn, y2mn or z2mn, and y2sn or z2sn). Note that PWC between 158 y (or z) and other predictor variables (e.g., y4 or z4) after excluding y2 or z2 and their 159 equivalent derivative variables (i.e., noised variables or variables with 0) are also calculated.

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The related results are not shown because they are analogous to those in case of predictor 161 variable of y2 (or z2). 162 The merit of the artificial data is that we know the exact scale-and localized bivariate 163 relationships after the effect of excluding variables is removed. Theoretically, we expect (a) 164 PWC is 1 at scales corresponding to scale difference of excluding variables from predictor 165 variable, and 0 at other scales. For example, PWC between y and y2y4 after excluding the 166 effect of y4 is expected to be 1 at the scale of 8, which is the difference of y4 (32) from y2y4 167 (8 and 32), and 0 at other scales (e.g., 32); (b) PWC remains 1 at the second half of series 168 where spatial series is replaced by 0, and 0 at the first half of the original series. For example,

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PWC between y and y2 after excluding the effect of y2h0 is expected to be 0 and 1 at the first     PWCsig decreases from 0.94 (y2sn) to 0.93 (y2mn) and 0.89 (y2wn) when progressively more 235 noise is added (Fig. 1a). If we exclude the predictor variable y2 itself, there are, as we expect, 236 no correlations between y and y2 (Fig. 1a). For the non-stationary case, similar results are 237 obtained (Fig. 1a). The only difference is that the scales with significant PWC values change 238 with location, as is found for MWC (Hu and Si, 2016).   corresponding to the predictor variable (Fig. 2a). For the stationary case, after excluding 249 the effect of y2h0, the PWC values are close to 1 (0.98) and 0 in the second and first half of 250 the data series, respectively, at the dimensionless scale of 8 (Fig. 2a). Similar results are 251 observed for the non-stationary case (Fig. 2a). This is anticipated because removing series  variable y2 (or z2) (Fig. 3), which is identical to the PWC between y (or z) and y2 (or z2) after 269 https://doi.org/10.5194/hess-2020-315 Preprint. Discussion started: 7 August 2020 c Author(s) 2020. CC BY 4.0 License. excluding the effect of variable y4 (or z4) (Fig. 1). After both predictor variables y2 and y4

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The PWC shows that SWC was not correlated with elevation after eliminating the effect of 377 SOC or depth to CaCO3 (Fig. 6). By contrast, after the removal of the elevation's effect,

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SWC was significantly correlated with SOC at scales of 36-144 m in the first half of the 379 transect and significantly correlated with depth to CaCO3 layer at large scales (>100 m) 380 across the transect (Fig. 6). There were little correlations between SWC and wetness index 381 after eliminating the effect of elevation (Fig. 6). Therefore, the influences of elevation and 382 wetness index on SWC in spring might have been taken into account by SOC and depth to 383 CaCO3 layer. Although elevation and wetness index are important drivers of snowmelt run-384 off in spring (Hu et al., 2017), they did not contribute any more to explaining SWC 385 variations than SOC or depth to CaCO3 layer did. The same holds for bulk density and sand 386 content whose influences on SWC were also limited after eliminating the effect of SOC 387 (Fig. 6). This was because SOC was negatively correlated with sand content at medium