1. Introduction
Soil moisture has been identified as a crucial component of the earth system, in particular owing to its role as a link between the water, energy, and carbon cycles (Seneviratne et al. 2010). Observations of its temporal and spatial dynamics have great value and potential in applications as diverse as streamflow prediction (Brocca et al. 2012), crop monitoring (de Wit and van Diepen 2007), weather forecasting (Scipal et al. 2008a), and the mapping of disease vector abundance (Chuang et al. 2012). One approach to mapping soil moisture is based on land surface models that are driven by or coupled with atmospheric models (Koster and Milly 1997; Koster et al. 2009; Balsamo et al. 2015). Satellite-based remote sensing products provide independent estimates of surface soil moisture (Kerr et al. 2012; Wagner et al. 2013; Dorigo et al. 2015), as do in situ probes, which are often considered to be the most accurate method (Seneviratne et al. 2010; Crow et al. 2012). However, observational networks are sparse, and these measurements furthermore typically refer to much smaller spatial scales. The spatial resolution is also a common source of disparity between different models and remote sensing products (Wagner et al. 2007; Western et al. 2002). Additional inherent differences between soil moisture products have also been attributed to, for example, different parameterizations, variable sampling depths, the impact of vegetation, or instrument measurement noise (Beven 2001; Crow et al. 2012; Famiglietti et al. 2008; Mialon et al. 2015). The discrepancies with respect to the underlying “true” soil moisture are commonly partitioned into systematic and random deviations (Entekhabi et al. 2010; Yilmaz and Crow 2013). Both have to be taken into account when feeding observations into a model or when combining different datasets, for example, in data assimilation for streamflow prediction or weather forecasting (e.g., Reichle and Koster 2004; Crow and van den Berg 2010; Yilmaz et al. 2012).
The triple collocation (TC) technique can be applied to three such products to yield consistent estimates of the variance of the random errors (Stoffelen 1998; Zwieback et al. 2012c; Su et al. 2014). It can do so without declaring one of the products to be the truth, that is to say, free of errors (Caires and Sterl 2003; Dorigo et al. 2010). The three products do not have to be calibrated with respect to one another: it is possible to account for differences in the mean (additive biases between the products) and in the magnitude (multiplicative biases or different sensitivities to the underlying soil moisture; Yilmaz and Crow 2013; McColl et al. 2014; Su et al. 2014). By compensating for additive and multiplicative biases, TC essentially assumes that the systematic components of the products are linearly related. Besides linearity, the TC estimation of the error variance is based on additional assumptions, in particular that the random errors are uncorrelated with each other and with the true soil moisture. The validity of these two assumptions has recently been questioned by Yilmaz and Crow (2014). Furthermore, the characteristics of the soil moisture products, and hence the relationships between the products, should not change over time, for example, seasonally. However, soil moisture datasets often differ in their seasonal cycle (Drusch et al. 2005), so that temporal anomalies are frequently analyzed rather than the standard products themselves (Brocca et al. 2011; Dorigo et al. 2010). Several studies have suggested extensions to TC that can account for complex temporal changes at various time scales (e.g., Loew and Schlenz 2011; Zwieback et al. 2013; Su and Ryu 2015). By contrast, the assumption of a linear relation between the datasets, that is, that—noise notwithstanding—they only differ by an offset and a different scaling factor, has not been addressed explicitly. The cumulative distribution function (CDF) matching procedure can partially account for them, but it is limited in that it does not explicitly consider errors in the soil moisture products (Yilmaz and Crow 2014).
Evidence for such nonlinear relations between different products has been found by, for example, Mittelbach et al. (2011), Drusch et al. (2005), De Lannoy et al. (2007a), and de Rosnay et al. (2009). For instance, Mittelbach et al. (2011) observed a declining sensitivity of an impedance probe as the soil became increasingly wet. De Rosnay et al. (2009) also identified a similar saturation effect when comparing areal soil moisture with point measurements. The associated discrepancy in the spatial scale may also have contributed. When upscaling point- to field-scale measurements, De Lannoy et al. (2007a) noted that a nonlinear method (CDF matching) was more accurate than a linear mapping. The gap in spatial scale was also a possible contributing factor to the nonlinear relations between in situ and remotely sensed soil moisture found by Drusch et al. (2005), in addition to inherent differences between these techniques. Many of the other previously identified origins of additive and multiplicative biases, such as differences in the sampling depth and the parameterization of the retrieval method or hydrologic model, may also contribute to nonlinear relations (De Ridder 2003; Koster et al. 2009).
Such nonlinear relations potentially impact common error metrics such as the correlation coefficient or the estimates obtained by the TC technique (Gruber et al. 2016). Their influence on TC error estimates of satellite soil moisture products has not been studied in detail before. We want to address this open question using both linear TC analysis and nonlinear techniques: the established nonparametric CDF matching technique as well as two new extensions of TC that can handle quadratic effects. The first one views TC as an instance of factor analysis (FA), a statistical technique that is widely applied in the social sciences and psychology (Wall and Amemiya 2007). The rationale of FA is to express observed quantities (the soil moisture products) as noisy linear measurements of a limited number of latent variables called factors (the underlying soil moisture). Nonlinear extensions of FA are usually based on additional assumptions: in our case we require that both the underlying soil moisture and the errors follow normal distributions. The second method that we propose extends the regression approach suggested by Scipal et al. (2008b) but is restricted to small quadratic nonlinearities. We analyze the accuracy and the robustness of these techniques in a simulation study. The accuracy in the presence of quadratic nonlinearities is assessed by comparing the simulation results with the known underlying error magnitudes as the degree of quadratic nonlinearity varies. In particular, this will allow us to see how a violation of the linearity assumption can impact the error estimates by the linear TC technique, for example, what magnitude and sign a possible bias can have. We furthermore analyze the robustness of the methods with respect to violations of their assumptions (e.g., normality and model misspecifications) using dedicated scenarios in which these assumptions are violated. Subsequently, we compare the methods in a small case study, where we assess a coarse-scale remote sensing product [Advanced Scatterometer (ASCAT)] by comparing it to a coarse-scale model (ERA-Interim/Land) and around 100 in situ probes of the U.S. Climate Reference Network (USCRN) over the contiguous United States. Such a combination of datasets is commonly employed in TC studies in order to characterize the uncertainty of the remotely sensed soil moisture product (e.g., Dorigo et al. 2015; Loew and Schlenz 2011): here, we focus on the sensitivity of the ASCAT error estimates to the choice of method, in particular whether it accounts for linear or quadratic relations. In contrast to the simulations, the underlying error magnitudes are not known, but the differences between the linear and the quadratic methods may give an indication of the impact of quadratic nonlinearities on TC error characterization. We apply these methods to both the standard products and the temporal anomalies. As soil moisture commonly exhibits a pronounced seasonal variation and as discrepancies in the climatology are frequently a major source of difference between different soil moisture datasets (Brocca et al. 2011), the removal of the seasonal signal when forming the temporal anomalies may potentially account for nonlinear relations induced by the mismatched climatological components. We thus hypothesize that the impact of nonlinearities on the error estimates of the temporal anomalies is smaller. Taken together, the simulations and the case study will provide a first insight into the practical relevance of nonlinear, especially quadratic, relations in TC error characterization of soil moisture datasets (Gruber et al. 2013; Su et al. 2014; Dorigo et al. 2015).
2. Triple collocation
Even though TC is commonly applied to time series of measurements, the temporal character of the data is only rarely accounted for explicitly (Zwieback et al. 2013; Su and Ryu 2015). We will also neglect the temporal character of the data, that is, the instances of the time series will essentially be treated as independent samples. In particular, this implies that the error terms are not autocorrelated and that the error variances and the offsets between the products (e.g., an additive bias) do not change.
For such matched products, the TC technique can yield unbiased and consistent estimates of the error variances 
errors have zero mean,
they are mutually uncorrelated, and
they are not correlated with the underlying soil moisture signal (i.e., T). If the latter is treated as a deterministic quantity, the error variances must not depend on T.
One way to estimate this error variance using TC was introduced by Stoffelen (1998). The method also provides estimates of the scaling factors with respect to a chosen reference product i, for which 
Rather than relying on the IV rescaling, Scipal et al. (2008b) suggested a similar approach that is based on total least squares (TLS). This regression technique yields estimates of the factors 
These two methods provide consistent estimates of the scaling factors 
3. Potential methods for tackling nonlinear relations
a. CDF matching
The idea of the CDF matching approach is to transform a soil moisture dataset so that its marginal distribution matches or approximately matches that of a reference product during the study period (Reichle and Koster 2004). Thus, a bias or a discrepancy in the dynamic range can be removed, and also the higher-order moments become identical (Gao et al. 2007). There are two general approaches to matching soil moisture products based on their distribution. The first one applies a mapping (or scaling) to one dataset so that the lower-order moments (e.g., the first and second) of the results equal those of the reference (Brocca et al. 2013; Yilmaz and Crow 2013). The alternative approach employed here operates directly on the empirical cumulative distribution functions of the reference product X and the product to be matched Y. Following Drusch et al. (2005) and Brocca et al. (2011), the two datasets are sorted and then the difference of the corresponding elements of the two ranked datasets computed. Subsequently, a polynomial in Y is fitted to the differences. This polynomial provides a correction term to Y so that its CDF approximately matches that of X. We use a polynomial of degree 5, which is considered to provide an adequate balance between the number of fitted parameters and model generality (Mahfouf 2010; Brocca et al. 2011, 2013).
This general method does not yield consistent estimates of the rescaling relations, which Yilmaz and Crow (2013) deemed detrimental for data assimilation studies. Negative effects also occur when estimating the error variances in standard linear triple collocation analysis of section 2 (see the treatment in section S1 of the supplemental material).
b. Factor analysis: EM algorithm
c. TLS extension
An alternative approach to incorporating nonlinear parametric functions may be to extend the error-in-variables (or TLS) regression approach employed by Scipal et al. (2008b) in the linear case. Their approach consists of three steps. Step 1 is the TLS regression, which provides estimates of the scaling factors 
The regression analysis of step 1 is also applicable to higher-order polynomials if the mean map of the reference product remains linear, for example, using the efficient algorithm by Boggs et al. (1987). However, the rescaling in step 2 of the original approach cannot be extended so easily. First, the inverse mapping from Y to its scaled version 
d. Method comparisons
Overview of the methods mk employed for error characterization, including the section where they are described and a summary of their assumptions.
4. Synthetic case study
a. Scenarios
The additional scenarios are distinguished from this base case as follows:
scenario linear (L): the quadratic terms in products Y and Z are set to zero, that is, μ = 0;
scenario quadratic 2 (Q2): the quadratic terms are twice as large, that is, μ = 0.2;
scenario quadratic 4 (Q4): the quadratic terms are four times as large, that is, μ = 0.4;
scenario right skewed (RS): T is modeled by a skew normal distribution (Azzalini 2005), where the first- and second-order moments remain the same and the skewness is set to 0.75 (the maximum possible value being 1.0);
scenario heavy tails (HT): both T and the noise terms have heavy tails, that is, a larger kurtosis, and they follow a Student’s t distribution with the same first and second moments as in
scenario bimodal (BM): T follows a bimodal mixture of Gaussians distribution (MacKay 2003), consisting of two components with equal weight and mean ±0.5 (maximum possible value being 1.0);
scenario truncated Gaussian (TG): T follows a truncated Gaussian distribution whose extent is limited to plus or minus one standard deviation, which is set to 1.85;
scenario low noise (LN): all mean maps are multiplied by 4, thus increasing the SNR as the error variance is kept constant;
scenario high noise (HN): all mean maps are divided by 4, thus reducing the SNR as the error variance is kept constant; and
quadratic 2 with switched mean maps (Q2s): same as Q2, but EM-Q and TLS-Q use quadratic mean maps for X and a linear mean map for Y.
b. Results and discussion
For these scenarios, the distributions of the RMSE estimates obtained using the methods of Table 1 are summarized in Figs. 1 and 2. For the linear scenario (i.e., L), all methods typically achieve comparable and satisfactory results, with median biases typically below 5%. In particular, the differences in both median and spread between the linear (TC and EM-L) and the quadratic (EM-Q and TLS-Q) methods are small compared to the variability of the estimates. Only for product Y are there noticeable (≈8%) median biases for methods TLS-Q and CDF.
Distribution of the RMSE (m3 m−3) estimates obtained with five different methods from Table 1 (distinguished by their color; TLS-L is not shown as the results are visually indistinguishable from TC) for four scenarios with increasing μ: L (μ = 0), nonlinear N (quadratic relation with μ = 0.1), and Q2 and Q4 (with μ = 0.2 and 0.4, respectively). The colored bars indicate the median of the 250 simulations, and the error bars span 95% of the distribution. The underlying true value is shown in red for reference. The annotation at the bottom of each bar gives the percentage of invalid estimates corresponding to negative variances.
Citation: Journal of Hydrometeorology 17, 6; 10.1175/JHM-D-15-0213.1
Distribution of the RMSE (m3 m−3) estimates obtained with five different methods from Table 1 (distinguished by their color; TLS-L is not shown as the results are visually indistinguishable from TC) for 11 scenarios: N (quadratic relation with μ = 0.1), RS, HT, BM, TG, LN, HN, Q2s, and L (μ = 0). The colored bars indicate the median of the 250 simulations, and the error bars span 95% of the distribution. The underlying true value is shown in red for reference. The annotation at the bottom of each bar gives the percentage of invalid estimates corresponding to negative variances.
Citation: Journal of Hydrometeorology 17, 6; 10.1175/JHM-D-15-0213.1
The differences between the methods are larger in the quadratic scenarios (Fig. 1). The median bias of the nonlinear EM method (i.e., EM-Q) barely (<5%) changes as the quadratic nonlinearity parameter (i.e., μ) increases. On the other hand, that of its linear counterpart (i.e., EM-L) increases with μ. Its sign and magnitude depend on the product. For product X, which is linearly related to the dimensionless anomaly T, it is negative and for Q4 it attains a value of more than 0.02 m3 m−3, corresponding to a relative magnitude of more than 40%. There is a positive bias for products Y and Z. Their magnitude increases nonlinearly with μ: for product Z it increases from 0.002 m3 m−3 at μ = 0.1 to 0.026 m3 m−3 at μ = 0.4. Comparable median biases also occur for the standard TC method. In addition, the underestimation for product X is concomitant with the occurrence of invalid negative variance estimates.
The TLS-Q method is also subject to biases in the nonlinear scenarios. These are not necessarily evident in the median bias but they can be related to the upward-skewed distribution of the estimates. Product Y is most affected by such skewed distributions. While the median bias increases to only less than 0.01 m3 m−3 in magnitude as μ increases to 0.4, the spread of the distribution more than doubles. Conversely, products X and Z are not characterized by a comparable increase in the spread of the estimates, but their median biases show an increasing trend with μ. However, they remain small compared to the linear methods (<40%). The median biases of the CDF matching method are comparable to those of TLS-Q in magnitude, that is, they are also typically at least a factor of 2 smaller than those of the linear methods. The CDF method is also affected by large spreads with upward-skewed distributions of the estimates. This effect is most pronounced for scenario Q4 for product X and tends to increase with μ for all products.
The CDF matching method is particularly affected by deviations from normality (Fig. 2). In the HT scenario, its estimates are not robust (95% coverage interval exceeding twice the RMSE for all products), and invalid values occur with a frequency of 14%. The EM-Q method, which is explicitly based on the assumption of normality, is less affected by heavy tails and also robust to skewed, bimodal, or truncated Gaussian distributions. Its linear cousin (EM-L) and also the standard TC method, which makes no distributional assumptions, achieve comparable results, but in a few cases (e.g., product Z and HT) the median biases increase by more than 50% compared to the base scenario. Also the TLS-Q method is found to be affected by deviations from normality, even though it is not based on such an assumption. While it is robust to heavy-tailed, skewed, and bimodal distributions, the distribution of the estimates becomes wide and biased in the TG scenario, similar to scenarios Q2 and Q4.
The scenario with low noise (LN) points toward the limitations of the linear methods (EM-L and TC), which are subject to median biases exceeding 0.02 m3 m−3 and which produce invalid estimates in up to 71% of the cases. A possible reason for this phenomenon may be the nature of the TC estimator, in which a positive quantity (the error variance) is estimated using a difference. For a fixed signal magnitude, the number to be estimated approaches zero as the SNR increases, so that any violation of the assumptions (e.g., a nonlinear mean map) can have a relatively larger impact. For product Z, the linear methods overestimate the RMSE, and the magnitude of this positive median bias is comparable to the scenario Q4 in which the quadratic nonlinearity parameter (i.e., μ) is 4 times as big. While the performance of the EM-Q and TLS-Q methods remains comparable to the base scenario, that of CDF matching deteriorates appreciably as the 95% coverage interval for products X and Z exceeds twice the RMSE. By contrast, a high noise level (HN) affects the TLS-Q most strongly, which is characterized by wide (95% coverage more than twice the RMSE) and upward-skewed distributions. The susceptibility of TLS-based triple collocation to high noise levels is consistent with the findings by Boggs et al. (1988), who analyzed and described the impact of random noise on the TLS regression estimates.
The TLS-Q method is more susceptible to misspecified mean maps than EM-Q in the Q2s scenario, in which the data are simulated with X and Y having linear and quadratic mean maps, respectively, whereas the mean maps are switched in the parameterization of TLS-Q and EM-Q. The results of TLS-Q show a larger median bias than the linear methods for product Z (0.14 m3 m−3). The 95% coverage is furthermore comparable to the RMSE of products X and Z. Conversely, EM-Q appears to be robust to such a model misspecification with, for example, a median bias of <0.01 m3 m−3 for product Z.
Overall, the EM-Q method consistently achieves the most reliable results, both in the presence of quadratic nonlinearities of varying magnitude and cases where the normality assumption is violated. This indicates that deviations from normality may be not critical in many situations, consistent with previous related studies (Yalcin and Amemiya 2001). Its linear counterpart and also standard TC, on the other hand, are susceptible to deviations from linearity. Such nonlinearities are also seen to affect TLS-Q. While this method is designed to handle small μ ≪ 1, its bias and spread (especially for μ = 0.4) suggest that this limitation may already be problematic at 
5. Uncertainty analysis of the ASCAT soil moisture product over the contiguous United States
a. Datasets and methods
The remotely sensed soil moisture product (TUW 2015) is derived from observations of the ASCAT instruments onboard the Meteorological Operational (MetOp) satellite series (Wagner et al. 2013). These active microwave instruments operate at C band (wavelength of 5.7 cm). The Vienna University of Technology (TU Wien) Water Retrieval Package, version 5.5 (WARP 5.5), algorithm is a change detection approach, with which time series of the DoS of the soil are derived from the multiangular backscatter observations at a resolution of 25 km (Naeimi et al. 2009; Wagner et al. 2013). The spatial posting of the product is 12.5 km. As the change detection model linking satellite observations and soil moisture does not apply during frozen conditions or snowmelt (Zwieback et al. 2015), such observations are screened using the approach by Naeimi et al. (2012).
The USCRN by the National Oceanic and Atmospheric Administration/National Climatic Data Center (NOAA/NCDC) consists of more than 100 climate monitoring stations within the contiguous United States. It is part of the International Soil Moisture Network (ISMN; Dorigo et al. 2011). All sites are equipped with dielectric soil moisture probes (Stevens Hydra Probe II SDI-12) at several depths, of which we use the hourly observations of VWC made at 5 cm depth (Bell et al. 2013; NCDC 2015). The quality controls by Dorigo et al. (2013) were applied to these data to filter out gross errors.
The ERA-Interim/ERA-Land dataset (ECMWF 2015) is based on the European Centre for Medium-Range Weather Forecasts (ECMWF) land surface model forced by ERA-Interim atmospheric data, with precipitation corrected toward observations (Balsamo et al. 2015). The dataset has a spatial resolution of around 80 km and a temporal sampling interval of 3 h (Albergel et al. 2013). The soil is represented by four layers, of which we use the topmost one (0–7 cm depth). The volumetric soil moisture content has been found to reflect that of in situ measurements accurately, thus making it suitable for comparisons with remotely sensed observations (Balsamo et al. 2015; Dorigo et al. 2015; Albergel et al. 2013).
For the remote sensing and reanalysis datasets, the grid cell closest to the location of each in situ probe is considered. Owing to the irregular temporal sampling of the remote sensing product, the datasets are collocated by choosing the in situ and reanalysis product nearest in time (within a window of 3 h; Dorigo et al. 2010). Stations with less than 100 collocated observations during the study period (from 1 January 2011 to 31 December 2014) are discarded, yielding a total of 112 stations. All methods of Table 1 were applied to these temporally matched soil moisture time series [referred to as standard products (std)] and also to short-term temporal anomalies (anom) calculated by subtracting the mean within a time window of 35 days (Brocca et al. 2011). Temporal anomalies are typically defined with respect to a long-term climatology, but the reference with respect to a moving average that is employed here is commonly used when the data are of limited temporal extent (Albergel et al. 2009; Dorigo et al. 2015). For both standard products and temporal anomalies, we prescribed linear mean maps for the ASCAT product and quadratic mean maps for the remaining ones. When assessing the sensitivity of the ASCAT error estimates to the choice of estimation method, we focus on the relative impact on the RMSE and SNR using the RD and RSNR metric, respectively (see section 3d).
b. Results
The error magnitude (RMSE) estimates for the ASCAT product based on the nonlinear quadratic EM method typically differ by less than 10% from those based on linear methods. The comparisons between all the methods are summarized in terms of the RD and D for the ASCAT product in Fig. 3 and Table S1 (in the supplemental material), and in terms of the estimated RMSE in Table S2 (in the supplemental material). On average, the error estimates are larger when estimated by EM-Q than when estimated by EM-L or TC (for both the standard products and the temporal anomalies). The larger errors of EM-Q correspond to lower SNRs (see Fig. S2 in the supplemental material), but the corresponding impact on the SNRs is typically less than 1.5 dB RSNR. The difference between the nonlinear and linear methods is found to be related to the estimated mean nonlinearity parameter (i.e., μ; see Fig. 4a). Larger relative differences (
Box plots of the RD (ASCAT) between different pairs of methods, summarizing all the 112 USCRN soil moisture stations. The method pairs m1, m2 are distinguished by their color according to the legend. The results for the standard products and the temporal anomalies are reported in the left and right half of the figure, respectively. Each box plot summarizes the respective median by a horizontal line, the interquartile range (IQR) by a colored box (the whiskers span 2.5 times the IQR), and the remaining values by plus signs. The vertical axis is linear from −1 to 1 and logarithmic in the absolute value for greater magnitudes.
Citation: Journal of Hydrometeorology 17, 6; 10.1175/JHM-D-15-0213.1
Relation between estimates of μ (i.e., the mean of μERA and μin_situ) and method comparison error metrics observed in the case study. Observed relation between μ and RD between (a) EM-Q and EM-L and (b) EM-Q and TLS-Q. (c) The inconsistency of the estimates of μ (in situ product) obtained using EM-Q and TLS-Q and the RD between these two methods.
Citation: Journal of Hydrometeorology 17, 6; 10.1175/JHM-D-15-0213.1
In absolute terms, the difference D between the RMSE estimates by EM-Q and the linear methods is typically (50%) less than 0.4% DoS in magnitude for the ASCAT product (Fig. 5a and Table S1 in the supplemental material). Around 15% of the stations have D(EM-Q, EM-L) exceeding 2% DoS in magnitude: in all these cases, the ASCAT error estimates by EM-Q are larger than those by its linear counterpart (i.e., EM-L). However, they rarely correspond to large 
Maps of the conterminous United States showing the in situ stations and different error metrics pertaining to all three products. The colors correspond to D(EM-Q, EM-L) for the (a) ASCAT and (b) ERA-Land product. The associated color bar is shown at top, and it also pertains to (c) D(EM-Q, EM-L) for the in situ measurements. (d) The estimate of the EM-L ASCAT RMSE (% DoS), along with its color bar.
Citation: Journal of Hydrometeorology 17, 6; 10.1175/JHM-D-15-0213.1
Maps of the conterminous United States showing the in situ stations and different error metrics pertaining to the ASCAT product. The colors correspond to the relative differences (a) RD(EM-Q, EM-L) and (b) RD(EM-Q, TLS-Q), respectively. The associated color bar is shown at top, and it also pertains to (c) RD(EM-Q, CDF). (d) The estimate of the ASCAT RMSE (% DoS), along with its color bar.
Citation: Journal of Hydrometeorology 17, 6; 10.1175/JHM-D-15-0213.1
The comparisons of the second nonlinear method TLS-Q with the linear methods reveal similar values with respect to D and RD. For both EM-Q and TLS-Q the disagreements with CDF matching are prominent (e.g., outliers with RD ≫ 10%). These disagreements are also larger when comparing the EM-Q and TLS-Q with each other than when comparing either of them with EM-L and TLS-L, respectively.
While the impact of the relaxed linearity assumption on the estimated error magnitudes is predominantly smaller than 10%, there are also stations with larger relative discrepancies that indicate more pronounced nonlinearities, such as those arising from the apparent saturation of one product with respect to another. The geographical distribution of these stations in Fig. 6 (and Fig. S4 in the supplemental material for the SNRs) does not show a clear correspondence to climate zones or topography. A particularly large 
Scatterplots of observed ASCAT (standard product or anomaly) and in situ soil moisture for six different stations. Each panel also shows the relation between the expected value of these two products for a given anomaly T as inferred by EM-Q (orange), TLS-Q (blue), and TC (green). The RD(EM-Q, EM-L) and RD(EM-Q, TLS-Q) are annotated in the upper-left and upper-right corner of each panel, respectively, whereas the name and geographic coordinates are given underneath.
Citation: Journal of Hydrometeorology 17, 6; 10.1175/JHM-D-15-0213.1
The discrepancies between the error estimates of the nonlinear and linear EM methods are more pronounced for the standard products than for the temporal anomalies. For the Spokane station in Figs. 7b and 7c, for instance, the 
Comparison of standard products and temporal anomalies with respect to 
Citation: Journal of Hydrometeorology 17, 6; 10.1175/JHM-D-15-0213.1
6. Discussion
a. Influence of quadratic nonlinearities on the error estimates
The impact of introducing nonlinear relations in the form of quadratic mean maps on the estimated ASCAT error magnitudes seems to be limited, as the RD between EM-Q and EM-L rarely exceeds 15%. Such large values are restricted to cases when the quadratic nonlinearity is pronounced (μ > 0.1, Fig. 4a). However, such a pronounced quadratic nonlinearity does not necessarily lead to large RD (see Fig. 4a). In other words, a large deviation from linearity as expressed by the quadratic measure (i.e., μ) seems to be a necessary but not a sufficient condition for EM-L to diverge appreciably from EM-Q. Also, the simulation study indicates that nonlinearities are not sufficient for EM-L to cease to provide reliable error estimates, as this estimator and also the other linear methods perform well for product Y despite its quadratic mean map with μ = 0.4 in scenario Q4. In this simulation scenario, the underestimation of the RMSE by EM-L (compared to both the true value and the estimate by EM-Q) for product X is accompanied by an overestimation for product Z. A similar seemingly compensatory deviation between EM-Q and EM-L for two products occurs in the case study, where about 15% of the stations have large negative D(EM-Q, EM-L) > 2% DoS for the ASCAT product and large positive D(EM-Q, EM-L) > 1% VWC for the ERA-Land product. Taken together, these findings indicate that the error induced by neglecting nonlinear quadratic relations can be of either sign for any given product. However, owing to its focus on the magnitude of the deviations, this study has not addressed ways of statistically assessing whether a nonlinear model provides a better fit. Possible ways of testing the evidence for nonlinear relations may include bootstrapping or the analysis of the likelihood within the factor analysis model, which, however, can be sensitive to the assumptions on which they are based, for example, regarding autocorrelation or normality (MacKay 2003; Yalcin and Amemiya 2001).
In the case study, the linear methods are more similar to each other than to the quadratic ones. The discrepancies of the nonlinear EM-Q with the standard TC method and also with TLS-L are comparable to those with EM-L. As the EM-L method relies on the normality assumption, its close agreement with standard TC may indicate that this assumption is not a limiting factor in the case study. A comparatively small impact of violations of this assumption was also observed in the simulation study of Fig. 2 for EM-L and EM-Q.
In contrast to the EM methods, the TLS-based algorithms make no assumption of normality. The quadratic version (TLS-Q), however, does require that the deviations from linearity be small. The simulation study confirmed this limitation by showing that the deviations (especially the spread) increase with the degree of quadratic nonlinearity. In the case study, such direct comparisons are impossible as the truth is not known. However, the disagreement between TLS-Q and EM-Q was found to increase with the estimated quadratic nonlinearity parameter (i.e., μ; Fig. 4). The observed dependence on μ suggests that the assumption of small nonlinearities can be a limiting factor in practice. The scatterplots of Figs. 7c–e show cases where the quadratic nonlinearity parameter estimated by TLS-Q exceeds that obtained with EM-Q. In both cases the TLS-Q estimates of μ may be interpreted as too large by a human observer, that is, the fitted relations seem questionable. However, in Fig. 7f the relation obtained with TLS-Q appears more natural than those estimated by EM-L or EM-Q. The latter two cases correspond to low SNRs, suggesting that the estimation of the mean maps is less stable when the soil moisture products are not closely related. As the simulation study has also indicated, such a lack of covariability limits the applicability and accuracy of TLS (Boggs et al. 1988) and also of standard TC (Su et al. 2014). While the simulations do not reveal associated limitations of the methods based on factor analysis, the limited scope of the synthetic case study does not preclude such limitations. Conversely, it does indicate a lack of robustness of TLS-Q to the specification of which product is assumed to have a linear mean map. This is in contrast to the EM-Q model based on factor analysis, which also requires that one of the products have a linear mean map: however, the simulations in Fig. 2 indicate that it is less affected by such a model misspecification. The simulation does not address model misspecifications with respect to the functional form of the mean maps, whereas, for example, Fig. 7a indicates that more flexible functions such as cubic polynomials may be more appropriate in practice. The appropriate model specification is hence an important issue that deserves further investigation.
An alternative way to match the products is the nonparametric technique CDF matching. As opposed to the other methods, it only compares the marginal distributions of the soil moisture products, that is, it does not explicitly account for errors. As shown in Fig. S1 (in the supplemental material; results based on normal distributions and linear mean maps) it can also be applied when the SNR is very small (10−1), provided that the SNR of the different products are equal (Yilmaz and Crow 2013). This requirement is likely problematic in practice, as the SNRs vary between the products, thus inducing biases even in the linear case. The simulations of section 4 suggest furthermore that deviations from normality (e.g., heavy tails) can exacerbate the biases induced by CDF matching. These biases are found to be mainly due to the wide and asymmetric distribution of the estimates, which additional simulations indicate is not due to the length of the time series. The origin and nature of this phenomenon thus deserve closer scrutiny, which may also affect potential mitigation strategies, for example, the use of nonparametric approximation approaches to matching the two CDFs. Despite these drawbacks, CDF matching has the advantage that it does not require a specific parametric form, as opposed to EM-Q and TLS-Q.
All the estimation methods may furthermore be affected when the error magnitude is related to the soil moisture content. Several examples in Fig. 7 (namely, Figs. 7a,d) indicate that this can indeed be the case. For instance, in Sebring in Fig. 7a, the disagreement between ASCAT and the in situ data increases as the soil becomes wetter. This trend appears to be related to the inferred mean maps, which indicate a saturation of the sensitivity. For this particular example, the RMSE varies by at least a factor of 3 between dry and wet conditions. Yilmaz and Crow (2014) and Mittelbach et al. (2011) also observed a connection between the random errors and the underlying soil moisture. As the average soil moisture often varies seasonally, the dependence of the error magnitude on the soil water content may also contribute to seasonal variations of the RMSE, as previously observed by, for example, Loew and Schlenz (2011) and Zwieback et al. (2012b). Uncertainty assessments that account for seasonal variations may therefore mitigate the impact of such a dependence even if they do not account for it explicitly. As these dependences can be large (e.g., a factor of 3 increase in the RMSE), they may also be relevant in data assimilation or product merging. If these applications are to account for variable error magnitudes, appropriate error characterization methods such as the multiplicative approach by Alemohammad et al. (2015) will be required.
b. Standard products and temporal anomalies
One of our central objectives was the comparison of the impact of nonlinearities on the error estimates of the anomalies compared to those of the standard products. We find that 
c. Practical relevance
Overall, the results of the case study suggest that the impact of quadratic nonlinear relations on the error characterization of soil moisture products may be limited in practice. The relative differences between linear and nonlinear quadratic methods of typically <10% may be smaller than other sources of uncertainty. One such source is the limited sample size. Previous studies have limited the error estimation to time series exceeding 100 samples (see section 5). The relative uncertainty of the RMSE for this length was estimated by Zwieback et al. (2012c) to be around 20%; this value depends on the statistical distribution of the noise terms. Additional sources of uncertainty (e.g., spatial mismatch and temporal variability) have also been identified as critical (Gruber et al. 2013, 2016; Su et al. 2015). Whether they are more important than nonlinear relations will depend on the application and dataset. In our study we find that nonlinearities seem to be relevant for about 15% of the stations [as measured by 
Accounting for nonlinear relations may also improve the merging and downscaling of soil moisture products. Two methods that can do so are CDF matching and copulas (Liu et al. 2012; Leroux et al. 2014). These are both built on a probabilistic framework but they do not explicitly consider the uncertainties of these products. However, probability theory can also be applied to incorporate these uncertainties, as, for instance, in factor analysis model of section 3b. Within its probabilistic model of (4), the merging of two soil moisture products X and Y may consist of a conditioning operation. The conditional distribution of T, 
7. Conclusions
Nonlinear relations between the three soil moisture products whose error variances are to be estimated cannot be handled by standard triple collocation (TC) analyses. Our simulations show that quadratic relations between soil moisture products can induce biases of either sign and that they can also be associated with negative, nonphysical error variance estimates. Triple collocation estimation can also be applied after preprocessing by CDF matching, which can account for arbitrary nonlinear relations: however, it does not explicitly consider random errors. Our theoretical analyses and simulations indicate that the accuracy and robustness of CDF matching (bias and spread of the estimates) are sensitive to the noise level, the probability distributions (e.g., normal, skewed, or heavy tailed), and the degree of quadratic nonlinearity of the products.
We introduce two estimation methods than can account for quadratic nonlinearities: they are both parametric methods that assume one of the relations to be linear but allow the others to be quadratic. First, total least squares (TLS) as part of triple collocation is extended to quadratic relations; however, it can only deal with small nonlinearities. In the simulation study, this technique becomes increasingly prone to outliers as the degree of quadratic nonlinearity increases. The presence and prevalence of these outliers, which are also related to the noise level, induce a bias, as they correspond to an overestimation of the error variance. Second, we develop a probabilistic method based on the variational expectation–maximization (EM) algorithm, which can handle both quadratic and linear relations. It assumes both the errors and the underlying soil moisture to be normally distributed. Empirically, however, this method is found to be the most robust error estimation approach in the simulation study, that is, it is least affected by nonlinearities of varying degrees and also nonnormal distributions (e.g., skewed or heavy tailed).
Comparisons of the quadratic EM method with its linear counterpart and the standard TC approach in a test study (coarse-scale remotely sensed and reanalysis soil moisture, in situ measurements) reveal typical relative differences in the error estimates of less than 10%. The discrepancies with CDF matching and both the linear and nonlinear TLS methods are of similar magnitude. However, we also observe a number of cases where quadratic nonlinear relations had a larger impact on the error estimates, with the relative difference exceeding 15%. Such cases occur almost exclusively for the standard products rather than the temporal anomalies, indicating that error estimates of the latter are typically less affected by quadratic nonlinearities. The observed quadratic relations correspond to the saturation of the sensitivity of one of the products with respect to others. They have been attributed in previous studies to, for example, different spatial scales, the parameterization of the hydrologic or remote sensing retrieval methods, or physical limitations of the sensors. Such nonlinear relations are usually not considered explicitly in applications of the soil moisture datasets, including data assimilation and product merging. In certain cases, these may profit from the error characterization provided by the suggested methods or new developments that can account for error autocorrelation or different functional forms of the nonlinear relations (e.g., cubic or higher-order polynomials). The explicit consideration of nonlinear relations thus has the potential to not only provide a more comprehensive uncertainty characterization of soil moisture products but also to improve the use of these products in applications as diverse as weather and flood forecasting.
Acknowledgments
The authors are grateful to Alexandra Konings, Kaighin McColl, and an anonymous reviewer for their insightful comments and suggestions, and to Christa D. Peters-Lidard for her handling of the manuscript submission and reviews. They gratefully acknowledge support from the European Commission via the FP7 project Earth2Observe (http://www.earth2observe.eu/; Grant Agreement 603608) and from EUMETSAT via the H-SAF (http://hsaf.meteoam.it/) project. They further thank Emanuel Dutra, Clément Albergel, and Gianpaolo Balsamo from ECMWF and Matthias Langer from ZAMG for providing the ERA-Land data. The ASCAT data are provided within the H-SAF project (http://hsaf.meteoam.it/).
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These methods are implemented in the software package nlscaling (http://dx.doi.org/10.5281/zenodo.44383).