# Estimation of Penetration Depth from Soil Effective Temperature in Microwave Radiometry

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{b}or soil moisture data are retrieved. Hence errors may arise because the Cal/Val data are not correspondingly sampled, in other words, are not comparable to the satellite observations. Additionally, different satellite soil moisture products may have different sensing depths, as different frequencies are used. As such, the various satellite soil moisture products may lack consistency and generate ambiguity in Cal/Val and their applications [21].

_{eff}. In general, all current two-layer ${T}_{eff}$ schemes use a weighting function for the soil temperature between upper layer and deeper layer. Such weighting function can be a constant [29], a fitting function [30,31], or an exponential function [15,16]. The weighting function is supposed to reflect the impact of soil moisture on the soil effective temperature. However, there is no variable of depth contained in Choudhury’s [29], Wigneron’s [30] or Holmes’ [31] ${T}_{eff}$ schemes. As indicated by the integral scheme, the weighting function would be more representative if it considers the influence of both soil moisture and soil temperature. However, it is difficult to quantify its effect on soil effective temperature, because soil temperature also affects soil moisture (e.g., as in dielectric constant models). In other words, ${T}_{eff}$ is a weighted mean of the soil temperature along the vertical profile. Therefore, it must be ${T}_{\mathrm{min}}<{T}_{eff}<{T}_{\mathrm{max}}$ (if ${T}_{\mathrm{min}}\ne {T}_{\mathrm{max}}$, e.g., non-uniform profile which is always the case for a land surface subject to radiative heating and cooling). Considering the diurnal variation and a semi unbounded soil column, T

_{max}and ${T}_{\mathrm{min}}$ usually appear at the surface skin or the deep layer where the soil temperature is almost constant. When the above condition is satisfied the sample layer covers the variation of ${T}_{eff}$. This also means as the soil temperature profile is continuous, there must be a layer where its soil temperature equals to ${T}_{eff}$.

## 2. Theoretical Background

#### 2.1. Microwave Radiative Transfer Model

_{eff}is expressed in terms of soil physical temperature of different layers, usually of two layers as

_{eff}= w

_{1}T

_{1}+ w

_{2}T

_{2}

#### 2.2. Soil Effective Temperature

_{1}used in Equation (6). The physical meaning of $\mathsf{\Delta}{x}_{1}$ could be inferred from Equation (6) that ${T}_{1}$ matches the layer-averaged soil temperature integrated from the surface to the sampling depth $\mathsf{\Delta}{x}_{1}$, which is used for calculating $1-{e}^{-{\tau}_{1}}$. It is to note that $\mathsf{\Delta}{x}_{1}$ (i.e., the bulk sampling layer thickness) is different from $\mathsf{\Delta}{x}_{1s}$ (i.e., the exact installation depth). Therefore, the soil moisture and soil temperature detected at $\mathsf{\Delta}{x}_{1s}$ represents average values from surface to $\mathsf{\Delta}{x}_{1}$, so that $\mathsf{\Delta}{x}_{1s}$ will be called the representative depth for the first layer. The representative depth is computed from the known installation depth for soil moisture and soil temperature sensors and has no relation to the deeper layers below. Let ${\tau}_{1}=\mathsf{\Delta}{x}_{1}\cdot \frac{4\pi}{\lambda}\cdot \frac{{\epsilon}^{\u2033}}{2\sqrt{{\epsilon}^{\prime}}}$ (noting $\mathsf{\Delta}{\tau}_{i}={\tau}_{i}-{\tau}_{i-1},and{\tau}_{i-1}=0$ for the first layer). Since soil depth at ith layer can be expressed as ${x}_{i}={\displaystyle \sum _{j=1}^{i}\mathsf{\Delta}{x}_{j}}$, it follows ${\tau}_{i}={\displaystyle \sum _{j=1}^{i}\mathsf{\Delta}{\tau}_{j}}$. Hence, $\tau $ monotonically increases with soil column depth $x$. With $\left[\tau ,T\right]$ instead of $\left[x,T\right]$ we can compute the correlation coefficient ($cc$) along the profile.

#### 2.3. Penetration Depth

## 3. Method and Data

#### 3.1. Predigest of Wilheit’s ${T}_{eff}$ Scheme

#### 3.2. Characteristic Expression of ${T}_{eff}$

_{i}is the optical depth at ith layer. One ${\tau}_{i}$ value corresponds to only one physical soil depth for a certain soil temperature/moisture combination at any moment (${\tau}_{i}={\tau}_{i-1}+\mathsf{\Delta}{x}_{i}\frac{2\pi}{\lambda}\frac{{\epsilon}^{\u2033}}{\sqrt{{\epsilon}^{\prime}}}$). As such, we can deem soil temperature a function of $\tau $. Furthermore, since the soil depth is between $[0,+\infty )$ as is $\tau $, we can use ${1-{e}^{-\tau}|}_{0}^{\infty}$ to represent the variation between $\left[0,1\right]$.

_{deep}= 5, the contribution from deeper layer $\tau >5$ is ${e}^{-5}\approx 0.0067$. Therefore, the soil temperature below ${\tau}_{deep}$ has negligible impact on ${T}_{eff}$. In other words, it does not matter where exactly ${\tau}_{deep}$ is as long as it is deep enough while Equation (14) is valid. We suggest ${\tau}_{deep}\ge 5$. According to Equation (12), we can calculate the normalized soil temperature:

_{nor}≤ 1. Put Equation (14) in Lv’s scheme as

#### 3.3. In-Situ Data, MERRA-2 and SMAP

## 4. Results

^{3}cm

^{−3}and soil temperature of 0–60 °C, the penetration depth ranges from 3–70 cm for L-band. When the soil is very dry (i.e., soil moisture is less than 0.01 cm

^{3}cm

^{−3}), the penetration depth is the greatest. Generally, the penetration depth would be 12 cm for L-band at 0.3 cm

^{3}cm

^{−3}and 30 °C. If the soil is not so dry, the effect of soil temperature needs to be considered. For instance, in Figure 3, the penetration depth could be 11 cm when soil moisture is 0.55 cm

^{3}cm

^{−3}and soil temperature is 50 °C. Nevertheless, the same penetration depth is also associated with soil moisture of 0.2 cm

^{3}cm

^{−3}and soil temperature of 10 °C. However, an error of soil temperature both in measurement and model simulation larger than 10 °C is rare and soil moisture dominate the penetration depth especially for dry soil. In this case, although the calculation of the penetration depth strongly depends on the dielectric constant model but the difference can be ignored. If the layers configuration were too sparse, the estimation would not be so precise in practice as in Figure 3. This is partly the reason why a vertically dense soil moisture and soil temperature profile was mounted at the Maqu Center Site with dense layers (19 layers within the top one meter). Such intensive layering would greatly minimize the uncertainty introduced by the dielectric models.

^{3}cm

^{−3}, the penetration depth varies from 6 to 10 cm at the center site. The average penetration depth is about 9 cm for the time before August 25 and 7 cm for the rest. As can be seen the penetration depth is strongly correlated with soil moisture which explains the variation in the period. Meanwhile, the penetration depth has its diurnal changes (around 1 cm) and is affected dominantly by soil temperature.

_{eff}as the mid-level of soil temperature profile in terms of Equations (1), (5) and (12) which could represent the average soil temperature of the soil column. Similarly, the soil moisture detected by satellites are also supposed to be the average soil moisture of the soil column in the view of emission depth. With T

_{eff}computed from MERRA-2 and global soil moisture map acquired from SMAP, Figure 5 illustrates a global distribution of the penetration depth by Equation (10).

_{eff}is higher.

## 5. Discussion

_{eff}.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Geographical location of the Maqu network on the Tibetan Plateau. The background indicates the elevation from USGS 1 km topography and the border in black is where elevation >2500 m; (

**b**) The distribution of all sites at the Maqu network and the center site (ELBARA) located in the center; (

**c**) ELABRA; (

**d**) the detailed soil moisture and soil temperature profile.

**Figure 2.**(

**a**) precipitation; (

**b**) the time series of soil moisture and (

**c**) soil temperature profiles at Maqu Network Center Station; (

**d**) the installation configuration of 20 sensors.

**Figure 3.**Penetration depth at L band (1.4 GHz). The ranges of penetration depth (in centimeters) were shown as contour lines, depending on the soil moisture and soil temperature. Mironov’s dielectric constant model was used here for calculating the real and complex parts of dielectric constants.

**Figure 4.**The time series of the penetration depth (Blue) and correlation coefficient (Red) between the soil temperature at the penetration depth and the corresponding soil effective temperature at Maqu Center Station as computed from the soil temperature/moisture profiles between 6 August and 27 November 2016.

**Figure 5.**Global map of the penetration depth (PD) for SMAP with (

**a**) minimum at 6 a.m.; (

**b**) minimum at 6 p.m.; (

**c**) maximum at 6 a.m.; (

**d**) maximum at 6 p.m.; (

**e**) mean at 6 a.m.; (

**f**) mean at 6 p.m. Data used are SMAP soil moisture passive L3 product and the corresponding soil effective temperature calculated from MERRA-2 for 2016. The SMAP soil moisture and soil effective temperature are considered as the mid-level values for each pixel vertically.

**Figure 6.**Comparison of soil temperature at the penetration depth vs. soil effective temperature at Maqu Center Station. The absolute correlation coefficient ($\left|cc\right|$) divided the time series into two groups where $\left|cc\right|>0.8$ (

**a**) and $\left|cc\right|<0.8$ (

**b**). The bottom figure shows the daily distribution of the moment when correlation coefficient $\left|cc\right|>0.8$.

**Figure 7.**Comparison of soil effective temperature calculated by Wilheit’s integral scheme against soil temperature observed at Maqu Center Station: (

**a**) 2.5 cm; (

**b**) 10 cm; (

**c**) 40 cm observation and (

**d**) the penetration depth. Data are shown only when $\left|cc\right|>0.8$ and the dashed line is the regression line. The period is from 6 August to 27 November 2016.

Abbreviation | Definition | Unit | Expression |
---|---|---|---|

${T}_{eff}$ | soil effective temperature | K | Equations (5), (6), (12) and (13) |

${T}_{b}$ | brightness temperature | K | |

$\theta $ | soil moisture | Vol/Vol | |

${T}_{\mathrm{max}}$ | maximum soil temperature along soil temperature profile | K | |

${T}_{\mathrm{min}}$ | minimum soil temperature along soil temperature profile | K | |

${T}_{i}$ | soil temperature at $i$th layer | K | |

${w}_{i}$ | weighting function for ${T}_{eff}$ | - | Defined in [29,31,34] |

$\mathsf{\Delta}{x}_{i}$ | soil thickness at $i$th layer | m | |

${x}_{(i)}$ | soil depth (at $i$th layer) | m | ${x}_{i}={\displaystyle \sum _{j=1}^{i}\mathsf{\Delta}{x}_{j}}$ |

$\mathsf{\Delta}{\tau}_{i}$ | optical thickness at $i$th layer | m | $\mathsf{\Delta}{\tau}_{i}=\mathsf{\Delta}{x}_{i}\frac{2\pi}{\lambda}\frac{{\epsilon}^{\u2033}}{\sqrt{{\epsilon}^{\prime}}}=\mathsf{\Delta}{x}_{i}\cdot \alpha \left(x\right)$ |

${\tau}_{i(x)}$ | optical depth at $i$th layer (or corresponding to soil depth $x$) | m | ${\tau}_{i}={\displaystyle \sum _{j=1}^{i}\mathsf{\Delta}{\tau}_{j}}$ |

${T}_{nor}/{T}_{inor}$ | normalized soil temperature (at $i$th layer) | - | ${T}_{(i)nor}=\frac{{T}_{(i)}-{T}_{surf}}{{T}_{deep}-{T}_{surf}}$ |

${T}_{surf}$ | skin temperature | K | |

${T}_{deep}$ | soil temperature at deep layer that the soil temperature could be considered as constant | K | |

$a$ | Soil temperature gradient | $K/\tau $ | $a=dT/d\tau $ |

$\alpha \left(x\right)$ | attenuation parameter | - | $\alpha \left(x\right)=\frac{4\pi}{\lambda}{\epsilon}^{\u2033}\left(x\right)/2{\left[{\epsilon}^{\prime}\left(x\right)\right]}^{\frac{1}{2}}$ |

${\tau}_{deep}$ | $\tau $ deep enough that the soil temperature could be considered as constant | - | ${\tau}_{deep}\approx 5$ |

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**MDPI and ACS Style**

Lv, S.; Zeng, Y.; Wen, J.; Zhao, H.; Su, Z.
Estimation of Penetration Depth from Soil Effective Temperature in Microwave Radiometry. *Remote Sens.* **2018**, *10*, 519.
https://doi.org/10.3390/rs10040519

**AMA Style**

Lv S, Zeng Y, Wen J, Zhao H, Su Z.
Estimation of Penetration Depth from Soil Effective Temperature in Microwave Radiometry. *Remote Sensing*. 2018; 10(4):519.
https://doi.org/10.3390/rs10040519

**Chicago/Turabian Style**

Lv, Shaoning, Yijian Zeng, Jun Wen, Hong Zhao, and Zhongbo Su.
2018. "Estimation of Penetration Depth from Soil Effective Temperature in Microwave Radiometry" *Remote Sensing* 10, no. 4: 519.
https://doi.org/10.3390/rs10040519