The SMOS-Derived Soil Water EXtent and equivalent layer thickness facilitate determination of soil water resources

The assessment of water resources in soil is important in understanding the water cycle in the natural environment and the processes of water exchange between the soil and the atmosphere. The main objective of the study was to assess water resources (in 2010–2013) in the topsoil from satellite (SMOS) and in situ (ground) measurements using the SWEX_PD approach (Soil Water EXtent at Penetration Depth). The SWEX_PD is a result of multiplying soil moisture (SM) and radiation penetration depth (PD) for each pixel derived from the SMOS satellite. The PD, being a manifold of the wavelength λ0 equal to 21 cm, was determined from the weekly SMOS L2 measurement data based on the real and imaginary part of complex dielectric constant. The SWEX_PD data were compared with soil water resources (WR) calculated from the sum of components derived from multiplication of soil moisture (SM) and layer thickness in nine agrometeorological stations located along the eastern border of Poland. Each study site consisted of seven neighbouring Discrete Global Grid pixels (nodes spaced at 15 km) including the central ones with agrometeorological stations. The study area included different types of soils and land covers. The agreement between the water resources obtained from the SWEX_PD and ground measurements (WR) was quantified using classical statistics and Bland–Altman's plots. Calibrated Layer Thickness (CLT = dbias) from 8 to 28 cm was obtained with a low values of bias (close to zero), limits of agreements, and confidence intervals for all the SWEX_PD, depending on the pixel location. The results revealed that the use of the SWEX_PD for assessing soil water resources is the most reliable approach in the study area. Additionally, the data from Bland–Altman plots and the equation proposed in these studies allowed calculation of the Equivalent Layer Thickness (ELT = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{ei}^{SWEX}$$\end{document}deiSWEX), which corresponds to the water resources derived from the SMOS satellite at the same time as (SM) measurements performed in the agrometeorological stations. The ranges of the mean, standard deviation, minimum, maximum, and coefficient of variation (CV) of ELT among all pixels and stations were 8.28–28.7 cm, 3.27–12.66 cm, 3.03–10.87 cm, 19.23–94.97 cm, and 24.72–98.79%, respectively. The ranges of the characteristics depended on environmental conditions and their means were close to the values of the calibrated layer thickness. The impacts of soil texture, organic matter, vegetation, and their interactive effects on the differentiation and agreement of soil water resources obtained from SWEX_PD vs. data from ground measurements in the study area are discussed. Further studies are required to address the impact of the environmental factors to improve the assessment of soil water resources based on satellite SM products (retrievals).


Materials and methods
Study area. The study was conducted in 9 sites in the eastern part of Poland (Fig. 1). Each site is equipped with agrometeorological station measuring soil moisture (SM). The soil moisture data for 2010-2013 (obtained every 3 days) used in the study came from the SMOS satellite (SMUDP2 v. 551) and were provided over the ISEA-4H9 (icosahedral snyder equal area Earth fixed) grid referred to as the discrete global grid (DGG) 36 . The nodes were equally spaced at 15 km. Next, 7 DGG pixels per each site were chosen in a way that the central one (named S0) contained the agrometeorological station and the other 6 (named S1, S2, S3, S4, S5, S6) were bordering around. Real and imaginary part of complex dielectric constant data from SMOS were used to determine the penetration depth using the approach described below. Penetration depth. The information about soil near-surface layer water resources is required for scientific elucidation of the global water cycle. The idea presented in this paper was to create SWEX_PD (Soil Water EXtent at Penetration Depth) for representing the amount of sub-surface soil water, substituting the surface Soil Moisture (SM) given by SMOS SMUDP2 data files. The multiplying soil moisture (SM) and radiation penetration depth (PD) was first proposed by W. Marczewski to determine uncalibrated soil moisture resources from SMOS measurements 37 .
The Penetration Depth (PD) is derived from dielectric constants based on the Kirchhoff Approach (KA) 28 . The approach assumes that SMOS gathering Temperature Brightness (TB) of emitted radiation over the soil layer thickness PD facilitates not only polarimetric retrieval for SM but also estimation of PD with KA for incoherent radiation. TB is used for determination of the intensity of radiation, while the next aspect is its attenuation by the layer PD-a concept analogical to the optical thickness of soil (also determined by the water). It is believed that, in this way, the surface Soil Moisture may be coupled to the absolute measures of water mass under observation, in kilograms per a particular surface, and the volume of the layer on the PD thickness.
According to the Beer-Lambert law 28 , the intensity of an electromagnetic wave in a uniform media falls off exponentially along the propagation path as Scientific Reports | (2020) 10:18330 | https://doi.org/10.1038/s41598-020-75475-x www.nature.com/scientificreports/ where I 0 is the electromagnetic wave in the air, α is the attenuation constant. We define the Penetration Depth (PD) for a path of z as a depth at which the intensity of electromagnetic waves decreases e times, where the Euler number e ≈ 2.718 defines 1 Neper in attenuation terms. We arbitrarily define: At depth z = PD, the electromagnetic wave components (E, H) have 1/e of their initial value. Since the power of the α wave in a particular medium is proportional to the square of intensity, the power at Penetration Depth is 1/e 2 , which is 13.5% of its initial value. The Kirchhoff Approach requires equality of the attenuation for radiation waves propagated by emission from media in two directions, i.e. for outsourcing and incoming waves. The attenuation constant for an electromagnetic wave at a normal incidence angle is proportional to the imaginary part of the material's refractive index n. It may be also expressed by the wave-extinction index κ: where ω is the angular frequency of radiation, λ 0 is the wavelength, and c is the speed of light in vacuum. The complex refractive index n and complex dielectric constant ε equally represent the propagation and loss properties of the medium and (in non-ferromagnetic materials) are related by 28 : where ε 1 is the real part and ε 2 is the imaginary part of the dielectric constant and i is the imaginary unit. After some simple transformations, one can obtain the conversions:  where Dielect_Const is the complex dielectric constant from SMOS L2 data, and RE and IM denote the real and imaginary parts of this complex number, respectively. PD is expressed in λ 0 , which for SMOS is 21 cm.
Soil Water EXtent at Penetration Depth (SWEX_PD). The SWEX_PD was proposed for representing the topsoil water amount in relation to the Soil Moisture (SM) at a depth corresponding to the radiometric Penetration Depth (PD) observed by SMOS in SMUDP2. The SWEX_PD concept couples surface SMOS data with the hydrology of deeper soil layers and may be an answer to the need of information about water retention in terms of water mass (in tons per square kilometre). Uncalibrated "water resources in soil" in the time j step derived from the SMOS satellites were calculated using the SWEX_PD jk as the product of soil moisture (SM jk ) and radiation penetration depth (PD jk ) for each SMOS pixel (k = 0, 1, 2,…n): SWEX j = SM j × PD j SWEX j = SM j × PD j Therefore, SWEX_PD jk is proportional to the soil moisture derived from SMOS but only in PD jk (not applicable to water resources in hydrological terms). PD jk was expressed in a wavelength λ 0 equal to 21 cm and was determined on the basis of weekly SMOS L2 measurement data based on the dependence of the real and imaginary parts of the dielectric constant.
Statistical approach. The uncalibrated (unscaled) SWEX_PD jk water resources derived from the SMOS satellite were compared with the water resources (WR j0 ) derived from ground measurements. To this end, Bland-Altman plots 38 were adopted as shown below. It was hypothesized that water resources are equal when the average of the differences SWEX_PD jk -WR j0 , the so-called bias, are close to or equal to zero: and the dispersion of the differences is finite and lies within a well-defined scatter path in the limit of agreement (LoA) and confidence interval (CI), which were calculated from the equations as: LoA = bias ± 1.96 × s (s-standard deviation), CI for bias: CI = bias ± t × (s 2 n -1 ) 0.5 , and CI for LoA: CI = LoA ± t × (3s 2 n -1 ) 0.5 , where n is the number of data and t is the value of t distribution with n-1 degrees of freedom 39 . The difference between the SWEX_PD jk and (WR j0 ) derived water resources was adjusted according to the equation: Selection of the Calibrated Layer Thickness (CLT = d bias ) at which the bias was close to zero or equal to zero was carried out. Depending on the type of sensors installed in the agrometeorological stations two steps coarse Scientific Reports | (2020) 10:18330 | https://doi.org/10.1038/s41598-020-75475-x www.nature.com/scientificreports/ and accurate were used. The layer thicknesses in the coarse step for the first sensor in the configurations PR2 probe (1) and the ThetaProbe ML2x (2) were: d i = 5, 10, and 15 cm and for other sensors they were d i = 5 and 10 cm. For the configurations ThetaProbe ML2x (5 cm) and the PR2 probe (3) and TDR sensors (4) the layer thicknesses for the individual sensors were: d i = 5 and 10 cm. The layer thickness of 1 cm was taken in the accurate step that allows to minimize the differences in soil water resources between those from satellite and ground-based measurements. The coarse step included the sum of the products of soil moisture (SM i ) and the thickness of the layers given above, checking how much the bias differed from zero. When the bias was much less than zero, the values of d i were decreased by the accurate steps (1 cm) and chosen so that the bias was close to zero. When the bias was greater than zero, d i was increased by the coarse or accurate steps so that the bias was close to zero. LoA and CI were also calculated to estimate the spread of the analysed data, and the a and b parameters of the linear regression equation were determined (see example in Fig. 2, site Majdanek). Values obtained from the linear regression equation and the bias value (Eq. 11) at the intersection point are equal and correspond to the CLT.
Only points close to bias on the Bland-Altman plot correspond to the CLT and those much above or below bias have different layer thicknesses resulting in overestimated or underestimated water resources. To reduce the deviations from bias the layer thicknesses were calculated from the transformed regression equation (Eq. 12) so that the regression equation line overlap with bias line (Eq. 13). The layer thickness calculated in this way, referred to as ( d SWEX ei ), was named Equivalent Layer Thickness (ELT) that corresponds to water resources derived from the SMOS satellite.
where SWEX_PD * jk is the water resource at the intersection point of the regression equation and the bias lines.
Ground measurements in agrometeorological stations. The ranges of sand, silt, and clay in the ground SM measurement stations were 22-97, 1-73, and 2-6% ( Table 1). The Biebrza and Bubnów stations (sites) were partly covered by marsh soils containing up to 80% of organic matter. The amount of organic matter in some parts of the Białowieża site reached up to 52%. In the other stations, there was 1-3% of organic matter.

Results
Comparison of soil water resources from SWEX_PD with ground measurements WR. The use of both coarse and exact steps depending on the SM sensor types and their installation depths allowed estimation (with an accuracy of 1 cm) of soil depths at which the soil water resources from ground measurements (WR j0 ) were similar or equal to those from the SWEX_PD jk based on the satellite data, as quantified by the Bland-Altman plots. The statistics including the differences SWEX_PD jk -WR j0 , i.e. bias, LoA, CI, and the linear regression equation SWEX_PD jk -WR j0 vs. (SWEX_PD jk + WR j0 )/2 for pixel S0 are shown in the Bland-Altman plots (Fig. 2). The linear regression equation parameters (a, b) for all pixels (S0-S6) are given in Fig. 3 along with biases at which the (WR j0 ) values were similar or equal to those from the SWEX_PD jk . The majority of SWEX_PD jk − WR j0 vs. (SWEX_PD jk + WR j0 )/2 values were within the defined LoA, some were within CI, and only few were beyond the areas of both LoAs and CIs. Such an arrangement of the data and the bias being close to zero indicated that the effect of the random factor was not significant and that the Bland-Altman approach can be used for the statistical analysis. As can be seen in Fig. 2, the LoA values for pixels S0 with SM sensors were generally in the range ± 0.1. Similarly, the LoAs for the other pixels (S1-S6) in each site were within this range (data not shown). It is worth noting that the largest and smallest LoAs were calculated for the grassed (> ± 0.1) and arable areas (around ± 0.07), respectively, in the same pixel Felin S0 (Fig. 2). Similarly, the CI values for bias and agreement limits (LoAs) were larger in the Felin-grassed than Felin-arable soil. As the line of equality (zero difference) lies between the confidence interval limits of bias, the latter is not significant. However, the CI of LoAs reached values around 0.02 in the Felin-grassed soil. This indicates that the estimated water resources from these two methods are more compatible in the Felin-arable soil. As can be seen in Fig. 3, the bias values are small (close to zero), positive or negative, in all sites and pixels. The linear regression coefficients (a) of the relationship SWEX_PD jk -WR j0 vs. (SWEX_PD jk + WR j0 )/2 (in Bland-Altman plots) were mostly negative with respective values < 0.5, 0.5-1.0, and > 1 for 6, 33, and 26 cases. They had positive values (from 0.096 to 1.536) only in 3 pixels in the Białowieża site (S0, S2, S6) and 1 pixel both in Trzebieszów (S1) and Felin-arable (S4).
As shown in Fig. 4, the largest calibrated layer thickness derived from the SMOS (d bias ) satellite was found in all or most pixels belonging to the Wigry, Biebrza, and Trzebieszów stations (from 26 to 28 cm), and lower values were observed in Majdanek, Felin-arable, and Bubnów (from 12 to 14 cm). The larger calibrated layer thickness indicates that the same amount of water in a given soil is distributed over a larger depth. The smallest and the largest differentiation of the calibrated layer thickness (from 8 to 15 cm) between the pixels was found

Discussion
Penetration depth (satellite sensing depth) and resultant soil water resources at a particular time depend on the soil textural composition, which determines the water-holding capacity 19,30 . In general, the penetration depth increases and the water-holding capacity decreases successively in clay, silt, and sandy soils. In our study, the effect of soil texture on the calibrated layer thickness determined by SEWX_PD or soil water resources derived from the SMOS satellite was not explicit. For example, a positive effect of the sand fraction can be clearly seen by comparison of the calibrated layer thickness from 19 to 26 cm for agricultural soils containing 58-86% of sand (Cicibór, Trzebieszów and Janów sites) with that from 13 to 18 cm for soils containing 22-26% of sand (and 68-73% of silt) (Majdanek and Felin stations- Table 1, Fig. 4). However, a negative effect of sand was observed while comparing the lower calibrated layer thickness in Biebrza (26 cm) with that in Wigry (28 cm) with the respective sand content of 97 and 87%. The lower calibrated layer thickness "seen" in the Biebrza station may result from the presence of organic matter-rich marshes (up to 80%), which retain much more water than inherently permeable sandy soils 40 . Also in the Bubnów site, the "seen" calibrated layer thickness was relatively small (10-12 cm) despite the relatively high content of sand (83%). Bubnów, similarly to Biebrza and Białowieża (52%), also had large amounts of organic matter in soil reaching up to 80%.
Our results showed the largest ranges of the 4-year calibrated layer thickness (CLT = d bias ) derived from the SMOS satellite (8-15 cm), equivalent layer thickness (ELT = d SWEX ei ) (from 8.3 to 16.09 cm), and coefficient of variation (CV) (from 40.5 to 99%) (Fig. 4) between the pixels in the Białowieża site. This site is situated in Białowieża National Park on the territory of Poland and Belarus with well-preserved forests and biological diversity, including the rare European bison. The area within the station is highly heterogeneous due to the spatial variation of soil organic matter content, textural composition, wetness, and vegetation. The area offers a diversity of conditions that influence the SMOS penetration depth retrievals. The analysis of the satellite-based pictures has demonstrated that the differences in the "seen" soil water resources (similarly, the differences in the equivalent layer thickness) between the pixels can be influenced by the type of predominant trees, i.e. either deciduous or coniferous. This can be illustrated by comparison of the relatively high CLT of 15 cm in pixel S5 covered predominantly by a coniferous forest with that of 8 cm in pixels S0 and S6 with predominance of broadleaved forests (Figs. 1 and 4). Broad-leaved trees vs. coniferous trees have a larger surface leaf area and (thus) water content, which may have a masking effect on the water "seen" in the soil 17 due to the presence of hydrogen stored in the vegetation affecting the counts of neutrons registered by the sensor 19 . The slight impact of coniferous trees can be supported by the same equivalent layer thickness "seen" (28 cm) in all pixels within the Wigry site consisting of four pixels (S0, S1, and S4-S6) used mostly as arable lands and two pixels (S2 and S3) covered by Table 1. Data on the study sites.  www.nature.com/scientificreports/ coniferous trees. (Fig. 1) In connection with this, a recent study in the tropics on the Tibetan Plateau revealed 11 that dense vegetation and associated small SMOS/SMAP-derived penetration depth need to be calibrated in contrast to limited or no vegetation in the dessert area exhibiting large penetration depth. El Hajj et al. 14 reported that the L bands (in HH polarization) penetrated a well-developed canopy cover of wheat and grasslands at the Normalized Difference Vegetation Index (NDVI) > 0.7 (the backscattered L-HH is sensitive to soil moisture), whereas the penetration of the C-band into the canopy was limited for an NDVI < 0.7. The variability in the equivalent layer thickness derived from the SMOS satellite among the pixels in the Białowieża site within Białowieża National Park can be further affected by the remaining dead trees. The presence of water accumulated in dead trees (not included in the estimation of soil moisture derived from the SMOS satellite) may have resulted in the lowest water depth "seen" (8 cm) in the soil in the forested areas within pixels S0 and S6 (Fig. 1) of the station. This effect can be more influenced by fallen than standing trees because the former can accumulate water 41 . It is worth noting that the low equivalent layer thickness "seen" in unforested pixel S2 can be influenced by organic matter in the marshes (52%) occurring in this pixel (Fig. 1). We observed visually that organic matter-rich soils (S1, S4) were saturated with water for most of the year and their topsoil is moist or www.nature.com/scientificreports/ sometimes dry only in the summer. In pixel S2 (in Belarus), the low "seen" calibrated layer thickness can also be influenced by the high content of silt (to 40%) and clay (to 11%) in the soil cover and organic matter in marshes. The effect of vegetation on the "seen" calibrated layer thickness (d bias ) is also visible while comparing the grassed (vegetated) and bare (arable) on the same (loess) soil within the Felin station. The greater calibrated layer thickness derived from the SMOS satellite for the grassed than bare soil (18 vs. 16 cm) can be a result of water uptake by grasses and the associated decrease in the soil water content and dielectric constant. On the other www.nature.com/scientificreports/ hand, in spite of the water uptake in soil covered by vegetation vs. bare soil, the dielectric constant can be higher in the former due to the lower evaporation from the soil surface 30 . This indicates that the "seen" calibrated layer thickness is affected by interactions between soil texture and vegetation.
It is worth noting that the equivalent layer thicknesses (ELT) for the Majdanek and Felin sites varying from about 5 to 25 cm during the study period were comparable to "sensing depth" from 8 cm in March to 27 cm in July observed in the SMOSREX (Surface Monitoring Of the Soil Reservoir EXperiment) microwave measurement campaign with similar soil 31,42 . Considering different sensing depths in the SMOSREX revealed that the agreement in soil moisture between the model-predicted data with consideration of the brightness temperature in the L band and measured data was satisfactory at depth of 0-5 cm which is concurrent with global surface soil moisture provided by the SMOS satellite 23 . In the case of spatially variable soils, the compliance of soil moisture from satellite and ground measurements in soil profile can be improved by using combined active, passive microwave and optical methods, with different penetration depths in airborne flights 43 . On the other hand, the obtained real and imaginary dielectric constants from the measurements of the SMOS brightness temperature allow to determine penetration depth and soil moisture using available methods 23,28,42 . The SWEX_PD and the approach proposed in this paper allowed for the determination of the calibrated and equivalent layer thicknesses which are consistent with the sensing depths (8-27 cm) obtained in the SMOSREX measurement campaign 42 and justify the use of SMOS SM data in our study. More detailed investigations are needed to address these interactions and enhance the usability of the satellite SM products (retrievals).

Summary and conclusions
The following findings were shown in this study: 1. The SWEX_PD was proposed for the first time to assess soil water resources based on 4-year SMOS satellite data and to determine the equivalent layer thickness derived from the SMOS satellite. The assessments were verified in nine sites under different soils and land cover (vegetation) in Poland. 2. The Bland-Altman plots including the bias, limits of agreements, and confidence intervals showed that the effect of a random factor was not significant and indicated that the soil water resources and equivalent layer thickness values were estimated with satisfactory accuracy. 3. Depending on the site conditions, the values of the equivalent layer thickness varied from 8.3 to 28.7 cm.
This variation was mostly attributed to differences in soil texture, organic matter, and vegetation and interactions between them. It was observed that the negative effect of the sand content on equivalent layer thickness values was masked (or compensated for) by the abundant organic matter in marsh soils and broad-leaved forest cover. The higher values of the equivalent layer thickness for the grassed than bare silt (loess) soil were ascribed to plant water absorption in the former. 4. Highly heterogeneous conditions in terms of soil organic matter, textural composition, wetness, and vegetation in one site situated within Białowieża National Park were reflected in a wide range of equivalent layer thickness values (8.26-16.09 cm) and spatial variability of the soil water resources (CV < 98.8%). 5. Further studies are intended to address the impact of the soil conditions and land cover more comprehensively to improve the assessment of soil water resources using satellite retrievals. Table 2. Ranges of the statistical parameters of equivalent layer thickness (ELT) and calibrated layer thickness (dbias) for data from all pixels in each station and from all pixels and stations. Explanations: min minimum, max maximum, SD standard deviation, and CV (%) coefficient of variation.