Study on soil moisture estimation using a three-frequency combination of observations integrated with robust estimation and machine learning

Geometric model of single ground reflection signal

Ground-based geodetic receivers for GNSS not only detect signals directly transmitted by satellites but also pick up signals bounced off the ground. This leads to interference at the receiver antenna’s center, resulting in multipath errors that are crucial in GNSS observations. These reflected signals mainly come from the initial Fresnel reflection area, particularly those undergoing specular reflection, which is the most impactful. For simplicity in theoretical analysis, we focus solely on the multipath effect caused by specular reflection. Figure 2 illustrates the near-surface forward reflection geometry model.

In Fig. 2, the RHCP antenna captures both the direct signal from and the ground-reflected signal. $\:{N}_{1}$ and $\:{N}_{2}$ correspond to the locations where signals undergo specular reflection from the two satellites. The GNSS measurement-type receiver antenna, labeled as $\:{N}_{3}$, is positioned at a vertical distance of $\:h$ units from the reflecting surface. Elevation angles for these satellites, observed from the antenna, are denoted by $\:{\theta\:}_{1}$ and $\:{\theta\:}_{2}$ respectively. Direct signals from these satellites received by the receiver antenna are designated as $\:{D}_{1}$ and $\:{D}_{2}$, while $\:{R}_{1}$ and $\:{R}_{2}$ represent the signals reflected from the two satellites respectively, after ground reflection.

Figure 2 illustrates the geometric arrangement, in relation to the direct signal, the reflected signal follows an extra path length, commonly referred to as the path difference⁴². This can be expressed mathematically as a function involving variables $\:h$ and $\:\theta\:$,

$$\:\varDelta\:\left(t\right)=2h{sin}\theta\:\left(t\right)$$

(1)

where, $\:h$ denotes the antenna height, while $\:\theta\:$ represents the incidence angle of the direct signal. Furthermore, in comparison to the direct signal, the relative time delay $\:\delta\:\left(t\right)$ and the excess phase of the reflected signal $\:\delta\:\phi\:\left(t\right)$, commonly referred to as the phase delay, are defined as³⁴,

$$\:\delta\:\left(t\right)=\varDelta\:\left(t\right)\text{/c}$$

(2)

$$\:\delta\:\phi\:\left(t\right)=2\pi\:\varDelta\:\left(t\right)/\lambda\:$$

(3)

where, Eq. (3) exclusively addresses geometric delay factors, neglecting phase contributions arising from the Fresnel reflection coefficient and antenna radiation pattern⁴². Here, $\:\lambda\:$ represents the signal wavelength, while $\:c$denotes the speed of light. When exclusively accounting for the specular reflection of signals on the object surface, the direct signal from the satellite, the reflected signal, and the interference signal formed by the superposition of the direct and reflected signals at the antenna can be expressed by the following Eqs¹³.,

$$\:Sd\left(t\right)=Ad{sin}(\psi\:\left(t\right))$$

(4)

$$\:Sm\left(t\right)=Am{sin}(\psi\:\left(t\right)+\delta\:\phi\:\left(t\right))$$

(5)

$$\:S\left(t\right)=Sd\left(t\right)+Sm\left(t\right)$$

(6)

where, $\:{A}_{d}$ represents the amplitude of the direct signal, while $\:{A}_{m}$ stands for the amplitude of the reflected signal, which can alternatively be denoted as $\:{A}_{m}=\alpha\:{A}_{d}$. Additionally, $\:\alpha\:$ denotes the amplitude attenuation factor (AAF), contingent on the reflectivity of the reflective surface media and antenna gain pattern³². Finally, the direct phase $\:\psi\:\left(t\right)$ (measured in radians) is expressed as:

$$\:\psi\:\left(t\right)=2\pi\:(N+\phi\:(t\left)\right)$$

(7)

where, $\:N$ and $\:\phi\:\left(t\right)$ represent the integer ambiguity and the direct phase, respectively, both measured in cycles.

Errors caused by multipath effects

When multipath phenomena occur near the GNSS geodetic receiver, denoted as $\:\beta\:\left(t\right)$ and $\:l\left(t\right)$, the carrier phase multipath error and pseudorange multipath error (measured in meters) can be mathematically expressed as follows⁴¹:

$$\:\beta\:\left(t\right)=\frac{\lambda\:}{2\pi\:}{{tan}}^{-1}\left(\frac{\alpha\:{sin}(2\pi\:\frac{2h}{\lambda\:}{sin}\theta\:\left(t\right))}{1+\alpha\:{cos}(2\pi\:\frac{2h}{\lambda\:}{sin}\theta\:\left(t\right))}\right)$$

(8)

$$\:l\left(t\right)=\frac{2h{sin}\theta\:\left(t\right)\alpha\:{cos}(2\pi\:\frac{2h}{\lambda\:}{sin}\theta\:\left(t\right))}{1+\alpha\:{cos}(2\pi\:\frac{2h}{\lambda\:}{sin}\theta\:\left(t\right))}$$

(9)

where, $\:\varDelta\:\left(t\right)$ and $\:{\delta\:}_{\phi\:}\left(t\right)$ represent the path difference and phase difference between reflected signals and direct signals, respectively, both functions of $\:{sin}\theta\:\left(t\right)$. Moreover, $\:\alpha\:={A}_{m}/{A}_{d}\ll\:1$, and $\:1+\alpha\:{cos}({\delta\:}_{\phi\:}\left(t\right))\approx\:1$⁵; additionally, when $\:x\ll\:1$,. Equations (8) and (9) can be further simplified as:

$$\:\beta\:\left(t\right)=\frac{\lambda\:}{2\pi\:}\alpha\:{sin}({\delta\:}_{\phi\:}\left(t\right))$$

(10)

$$\:l\left(t\right)=\alpha\:\varDelta\:\left(t\right){cos}({\delta\:}_{\phi\:}\left(t\right))$$

(11)

Considering that the soil moisture and GNSS satellite orbit coordinates are organized in a continuous time series, the variables mentioned earlier, denoted as ($\:h$, $\:l\left(t\right)$, $\:\theta\:\left(t\right)$, $\:\tau\:\left(t\right)$, $\:\varDelta\:\left(t\right)$, $\:{\delta\:}_{\phi\:}\left(t\right)$, $\:\alpha\:$), are all time-dependent. For simplicity, the subsequent discussion omits the time-related terms⁴³.

The proposed combination of observations

In GNSS positioning, when solely accounting for the impact of atmospheric delays, the observation equations for carrier phase $\:{\phi\:}_{i}^{s}$ (measured in cycles) and pseudorange $\:{P}_{i}^{s}$ (measured in meters) acquired by GNSS geodetic receivers at each epoch can be approximately formulated as follows⁴³:

$$\:{\lambda\:}_{i}{\phi\:}_{i}^{s}={d}^{s}+c({t}_{R}-{t}^{S})-{I}_{i}^{s}+{T}^{s}+{\lambda\:}_{i}{N}_{i}^{s}+{\beta\:}_{i}^{s}$$

(12)

$$\:{P}_{i}^{s}={d}_{i}^{s}+c({t}_{R}-{t}^{S})+{I}_{i}^{s}+{T}^{s}+{l}_{i}^{s}$$

(13)

In this scenario, the satellite identifier (PRN) is denoted by the superscript $\:s$ and the frequency band of the GNSS satellite signal is indicated by the subscript $\:i$. Geometric distance between the satellite and the receiver, typically unknown, is represented by the parameter $\:d$. Clock deviations of the receiver and the satellite are marked as $\:{t}_{R}$ and $\:{t}^{S}$ respectively, while the speed of light is denoted by $\:c$. Ionospheric delay, contingent upon the frequency band, is indicated by $\:{I}_{i}^{s}$, and tropospheric delay, introduced during signal propagation through the troposphere and dependent on atmospheric conditions along the satellite-receiver path but unrelated to the signal frequency band, is signified by $\:{T}^{s}$. Carrier phase and pseudorange multipath errors are represented by $\:{\beta\:}_{i}^{s}$ and $\:{l}_{i}^{s}$ respectively, stemming from multipath effects near the receiving station.

Linear combination of the three C/A codes pseudorange and the three L-frequency bands carrier phase are expressed as follows:

$$\:{\phi\:}_{\text{1,2},5}={k}_{1}{\lambda\:}_{1}{\phi\:}_{1}^{s}+{k}_{2}{\lambda\:}_{2}{\phi\:}_{2}^{s}+{k}_{5}{\lambda\:}_{5}{\phi\:}_{5}^{s}$$

(14)

$$\:{P}_{\text{1,2},5}={k}_{1}{\lambda\:}_{1}{P}_{1}^{s}+{k}_{2}{\lambda\:}_{2}{P}_{2}^{s}+{k}_{5}{\lambda\:}_{5}{P}_{5}^{s}$$

(15)

The ionosphere is formed by solar radiation ionizing a segment of the atmosphere, generating electrons that induce a delay in signal propagation, termed the ionospheric delay. This delay’s magnitude is mainly influenced by the frequency of the signals⁴⁴. Substituting Eqs. (12) and (13) into Eqs. (14) and (15) completely removes the ionospheric delay error and collapses to give:

$$\:{\phi\:}_{\text{1,2},5}={\beta\:}_{\text{1,2},5}+{N}_{\text{1,2},5}$$

(16)

$$\:{P}_{\text{1,2},5}={l}_{\text{1,2},5}$$

(17)

where,

$$\:{\beta\:}_{\text{1,2},5}={k}_{1}{\lambda\:}_{1}{\beta\:}_{1}^{s}+{k}_{2}{\lambda\:}_{2}{\beta\:}_{2}^{s}+{k}_{5}{\lambda\:}_{5}{\beta\:}_{5}^{s}$$

(18)

$$\:{l}_{\text{1,2},5}={k}_{1}{\lambda\:}_{1}{l}_{1}^{s}+{k}_{2}{\lambda\:}_{2}{l}_{2}^{s}+{k}_{5}{\lambda\:}_{5}{l}_{5}^{s}$$

(19)

$$\:{N}_{\text{1,2},5}={k}_{1}{\lambda\:}_{1}{N}_{1}^{s}+{k}_{2}{\lambda\:}_{2}{N}_{2}^{s}+{k}_{5}{\lambda\:}_{5}{N}_{5}^{s}$$

(20)

$$\:{k}_{1}={\lambda\:}_{5}^{2}-{\lambda\:}_{2}^{2},{k}_{2}={\lambda\:}_{1}^{2}-{\lambda\:}_{5}^{2},{k}_{1}={\lambda\:}_{2}^{2}-{\lambda\:}_{1}^{2}$$

(21)

After fixing the weekly jump, the whole week fuzzy degree combination term, represented by the second term on the right side of Eq. (16), can be viewed as a constant. Consequently, the numerical equivalence between the three-frequency multipath error and the linear combination of the three-frequency GNSS observations is established.

Compared to SNR-based GNSS-IR methods, the proposed TFPC and TFCPC methodologies simplify soil moisture retrieval by eliminating the need to separate the reflection component from direct signals. Unlike SNR methods requiring wide-elevation-angle data and primary frequency analysis, the phase-based methods exhibit better tolerance to unstable environmental conditions, potentially resulting in more robust outcomes and practicality. However, these methodologies introduce increased measurement noise from combined observations, thus necessitating robust detection and correction of outliers. Observations from Figs. 3 and 4 indicate that each series of multipath errors displays numerous minor fluctuations, mainly due to residual components of random measurement noise. Additionally, the overall fluctuations of both TFCPC and TFPC multipath errors decrease as the sine value of satellite elevation angle increases, indicating that multipath effects are more pronounced for satellite signals at lower elevation angles. This observation provides insight into intentionally choosing the satellite elevation angle for GNSS-IR inversion. Subsequent experiments focused on multipath errors within the 5°–30° low elevation angle range to optimize outcomes.

Generation of abnormal phases

Analyzing multi-path error signals in time series can obtain signal components and spectral relationships^46,47.Lomb–Scargle spectral analysis enables the determination of the peak frequencies and oscillation periods of combined multipath errors^45,48. The power spectral density (PSD) at the peak frequency experiences a noticeable reduction in the presence of measurement noise, as opposed to the noise-free scenario. Additionally, a slight deviation in the peak frequency occurs due to the influence of measurement noise³⁴. Typically, the maximum PSD acts as an indicator of the multipath error signal quality³³. Furthermore, the power spectral density (PSD) at the primary frequency in the Lomb–Scargle periodogram (LSP) should ideally be twice as high as the power of the noise or the second largest frequency²⁴. Therefore, the Lomb–Scargle spectral analysis provides a method to evaluate the integrity of multipath error signals. In the case of a specific reflecting surface, the satellite orbit should demonstrate a consistently stable singular frequency²⁵.

Due to the influence of random measurement noise in single-frequency pseudorange and carrier phase observations, high-quality Three-frequency combined signals may not always be present. The spectral analysis periodograms depicted in Figs. 5 and 6 illustrate TFCPC and TFPC multipath errors across various satellites. In specific subfigures, the dominant frequency power of the LSP is twice that of the secondary frequency power, indicating favorable combined multipath error characteristics. However, in other subplots, the dominant frequency power is less evident, potentially due to diverse influences such as slope direction, surface roughness, surrounding vegetation canopy, and random measurement noise affecting multipath signals from differently oriented satellite orbits. Over the experimental duration, the consistent occurrence of high-quality multipath errors cannot be assured.

Signals from combined GNSS satellite orbits impacted by noise lack clear frequencies, rendering multipath errors of inferior quality akin to observations containing coarse inaccuracies. Moreover, the traditional least squares method lacks robustness, particularly when exposed to multipath errors containing coarse errors. Feeding such erroneous data into the least squares analysis can lead to its degradation⁴⁹. Consequently, the computed results manifest as anomalous phase delays. Thus, the effective detection and correction of anomalous phases are crucial for improving the accuracy of GNSS-IR soil moisture estimation.

Robust estimation

When estimating soil moisture through GNSS-IR, the actual errors in observations typically adhere to a normal distribution. Outliers in delay phases can often be identified through integer multiples of the root mean square error. However, this method is susceptible to the impact of outliers, resulting in inaccurate detection results such as overlooking certain outliers or incorrectly identifying normal values as outliers, referred to as false negatives and false positives, respectively. To address this challenge, the study introduces an innovative outlier detection method utilizing minimum covariance determinant (MCD) robust estimation. The primary aim of this method is to construct a robust estimate of the covariance matrix using the Mahalanobis distance and iterative concepts. The robust Mahalanobis distance is iteratively computed and validated through the utilization of a Chi-square distribution. The methodology allows for the identification of anomalies within the dataset and assigns different weights to each detected anomaly⁵⁰.

From a dataset containing $\:n$ samples of known dimensionality, a random sample of size $\:m$ is selected. Generally, as the value of m decreases, the MCD robust estimation method demonstrates enhanced outlier handling capabilities. Yet, excessively small values of $\:m$ pose difficulties in distinguishing abnormal values from normal ones. Thus, the default value of $\:m$ is typically set to $\:0.75n$. Here, the initial sample mean and covariance matrix are derived from the mean and covariance matrix of the sample, represented by the following formula⁴²,

$$\:{u}_{1}=\frac{1}{h}{\sum\:}_{i=1}^{h}{X}_{i}$$

(22)

$$\:{S}_{1}=\frac{1}{h}{\sum\:}_{i=1}^{h}({X}_{i}-{u}_{i}\left)\right({X}_{i}-{u}_{i}{)}^{T}$$

(23)

Equations (18) and (19) are utilized to compute the Mahalanobis distance between the dataset and the mean of the chosen sample, as expressed below,

$$\:MD\left(i\right)=\sqrt{({X}_{i}-{u}_{i}{)}^{T}{S}_{1}^{-1}({X}_{i}-{u}_{i})}$$

(24)

$\:n$ Mahalanobis distances, as per Eq. (20), are computed and sorted in descending order. Next, $\:h$ sample data points with the lowest distances are selected. Using these samples, the mean estimate and covariance matrix estimate are recalculated. This process iterates until either $\:{det}({S}_{i-1})={det}({S}_{i})$ or $\:{det}({S}_{i})=0$ conditions are met. The determinant of the covariance matrix in this iterative process follows the subsequent relationship.

$$\:{det}({S}_{1})\ge\:{det}({S}_{2})\ge\:…\ge\:{det}({S}_{i-1})\ge\:{det}({S}_{i})$$

(25)

The iterative process described in Eq. (21) is shown to converge. Upon completion of the iteration, the mean and covariance matrix obtained represent the robust mean and robust covariance, respectively. These values substituted into the Mahalanobis distance equation allow for the computation of Mahalanobis distances for the dataset, following a Chi-square distribution with $\:p$ degrees of freedom. If Eq. (22) is satisfied, the sample point is classified as an outlier; otherwise, it is considered a normal value.

$$\:MD\left(i\right)>\sqrt{{\chi\:}_{p,\alpha\:}^{2}}$$

(26)

Employing MCD has proven effective in pinpointing and eliminating outlier positions. To maintain the temporal coherence of delay phases while preserving their accuracy, a moving average filter (MAF) is deployed. This approach, implemented using the built-in smooth function in MATLAB software, corrects aberrant phases. By executing these steps, a larger number of satellites are ensured to display high-quality delay phases, thus broadening the selection of satellites for subsequent experiments.

Error equation establishment and parameter solving

Inferring from Eqs. (1) and (3), it becomes apparent that both wave delay $\:\varDelta\:$ and delay phase $\:{\delta\:}_{\phi\:}$ are dependent on the antenna height h. Over a certain period, fluctuations in soil moisture levels result in varying antenna heights, thereby causing fluctuations in wave delay and delay phase. However, the correlation between soil moisture and delay phase trends is more consistent than that between soil moisture and antenna height. Additionally, considering microwave reflection mechanisms, changes in soil moisture can alter the soil’s dielectric constant and surface reflectivity, thereby affecting the variability of the amplitude fading factor. Consequently, the nonlinear observation equation is linearized to derive the error equation, allowing for the computation of the path difference, phase delay, and AAF, and assessing their ability to reflect soil moisture variability,

$$\:l+{V}_{l}={\alpha\:}_{0}{\varDelta\:}_{0}{cos}\delta\:{\phi\:}_{0}+{\varDelta\:}_{0}{cos}\delta\:{\phi\:}_{0}{V}_{\alpha\:}+{\alpha\:}_{0}{cos}\delta\:{\phi\:}_{0}{V}_{\varDelta\:}-{\alpha\:}_{0}{\varDelta\:}_{0}{sin}\delta\:{\phi\:}_{0}{V}_{\delta\:\phi\:}$$

(27)

$$\:\beta\:+{V}_{\beta\:}={\alpha\:}_{0}{sin}\delta\:{\phi\:}_{0}+{sin}\delta\:{\phi\:}_{0}{V}_{\alpha\:}+{\alpha\:}_{0}{cos}\delta\:{\phi\:}_{0}{V}_{\delta\:\phi\:}$$

(28)

where, observations $\:l$ and $\:\beta\:$ are readily calculated through Eqs. (14) and (15), and $\:{V}_{l}$ and $\:{V}_{\beta\:}$ denote the infinitesimal adjustments of the TFCPC and TFPC multipath errors, also known as the corrections to the observations $\:l$ and $\:\beta\:$. The $\:{\alpha\:}_{0}$ represents the initial value of the amplitude attenuation factor, and the amplitude is required to calculate. The direct signal amplitude usually defaults to 1, yet the reflected one is equal to the product of the antenna gain ratio between the positive and negative elevation angle and the media surface reflectivity (0.3–0.8). Since the primary focus of this paper is to explore the relationship between delay phases and soil moisture, the value is set to 0.3 when ignoring the influence of antenna gain.

Equation (1) enables deduction of the initial value of the path difference, denoted by $\:{\varDelta\:}_{0}$, based on the antenna height $\:h$ and satellite elevation angle $\:\theta\:$. The initial value of the delay phase, represented by $\:{\delta\:}_{{\phi\:}_{0}}$, can be computed using the wavelength $\:\lambda\:$ and path difference $\:{\varDelta\:}_{0}$ as previously indicated in Eq. (3). Correction terms $\:{V}_{\alpha\:}$, $\:{V}_{\varDelta\:}$, and $\:{V}_{\delta\:\phi\:}$ are applied to the corresponding estimated parameters, namely $\:\alpha\:$, $\Delta$, and $\:{\delta\:}_{\phi\:}$ (TFCPC: $\:{V}_{\alpha\:}$, $\:{V}_{\delta\:\phi\:}$; TFPC: $\:{V}_{\alpha\:}$, $\:{V}_{\varDelta\:}$, $\:{V}_{\delta\:\phi\:}$).

Considering Eqs. (8), (9), (16), and (17), it becomes clear that TFCPC and TFPC multipath errors rely on the satellite elevation angle $\:\theta\:$. In this study, we utilize the least squares method to estimate the characteristic parameters. Due to significant fluctuations in multipath errors across various elevation angles, utilizing an extended combined multipath error sequence for parameter estimation may not yield the most accurate parameter values. The excess multipath errors are treated as observations containing coarse inaccuracies, ensuring that the length of the multipath error series exceeds the number of parameters to be estimated, thereby ensuring a unique solution for the equation. Consequently, for parameter estimation in this investigation, we opt for a Three-frequency combined multipath error sequence with a length of 5. We assume that the path difference and delay phase remain constant over a short duration while disregarding minor fluctuations in soil moisture.

Hence, we selected a multipath error sequence with a length of 5, along with the corresponding sequence of satellite elevation angles. Through assigning uniform weights to each data point, indirect leveling error equations were constructed to ascertain the optimal estimated parameter values. It’s crucial to emphasize that when determining the initial values of the path difference and delay phase, the computed results derived from the initial value of the satellite elevation angle sequence should be adhered to.

In this research, we employed five combined multipath errors to construct an observation equation for indirect adjustment. In the TFPC method, the parameter $\:m$ was assigned the value 3, whereas in the TFCPC method, $\:m$ was set to 2. The number of error equations corresponds to the number of combined multipath errors. The estimation process for the characteristic parameters of a single satellite can be illustrated in the following matrix format:

$$\:\underset{5*1}{V}=\underset{5*m}{B}\:\underset{m*1}{x}\:-\underset{5*1}{L}$$

(29)

The construction of the matrices for each part of the error equation is described separately for the TFCPC and TFPC methods in the following sections,

$$\:\underset{5*1}{V}=[\:{V}_{1}\text{}{V}_{2}\text{}\cdot\:\cdot\:\cdot\:\:{V}_{5}\:{]}^{T}$$

(30)

$$\:\underset{5*2}{{B}_{\beta\:}}={\left[\begin{array}{c}\text{}\text{}\text{}\text{}{sin}(\delta\:{\phi\:}_{0}\left)\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}{sin}(\delta\:{\phi\:}_{0}\right)\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\cdot\:\cdot\:\cdot\:\text{}\text{}\text{}\text{}\text{}\text{}{sin}(\delta\:{\phi\:}_{0})\\\:\text{}\text{}{\alpha\:}_{0}{cos}(\delta\:{\phi\:}_{0}\left)\text{}\text{}\text{}{\alpha\:}_{0}{cos}(\delta\:{\phi\:}_{0}\right)\text{}\text{}\text{}\text{}\text{}\cdot\:\cdot\:\cdot\:\text{}\text{}\text{}\text{}{\alpha\:}_{0}{cos}(\delta\:{\phi\:}_{0})\end{array}\right]}^{T}$$

(31)

$$\:\underset{5*3}{{B}_{l}}={\left[\begin{array}{c}\text{}\text{}\text{}{{\Delta\:}}_{0}{cos}(\delta\:{\phi\:}_{0}\left)\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}{{\Delta\:}}_{0}{cos}(\delta\:{\phi\:}_{0}\right)\text{}\text{}\text{}\text{}\text{}\text{}\cdot\:\cdot\:\cdot\:\text{}\text{}\text{}\text{}\text{}\text{}{{\Delta\:}}_{0}{cos}(\delta\:{\phi\:}_{0})\\\:\:-{\alpha\:}_{0}{\varDelta\:}_{0}{sin}(\delta\:{\phi\:}_{0})\text{}\text{}\text{}-{\alpha\:}_{0}{\varDelta\:}_{0}{sin}(\delta\:{\phi\:}_{0})\text{}\text{}\text{}\cdot\:\cdot\:\cdot\:\text{}\text{}\text{}-{\alpha\:}_{0}{\varDelta\:}_{0}{sin}(\delta\:{\phi\:}_{0})\\\:\text{}\text{}\text{}{\alpha\:}_{0}{cos}(\delta\:{\phi\:}_{0}\left)\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}{\alpha\:}_{0}{cos}(\delta\:{\phi\:}_{0}\right)\text{}\text{}\text{}\text{}\text{}\text{}\text{}\cdot\:\cdot\:\cdot\:\text{}\text{}\text{}\text{}\text{}{\alpha\:}_{0}{cos}(\delta\:{\phi\:}_{0})\end{array}\right]}^{T}$$

(32)

$$\:\underset{2*1}{{x}_{\beta\:}}=[\:{x}_{\alpha\:}\text{}{x}_{\phi\:}{]}^{T}$$

(33)

$$\:\underset{3*1}{{x}_{l}}=[\:{x}_{\alpha\:}\text{}{x}_{\phi\:}\text{}{x}_{\varDelta\:}{]}^{T}$$

(34)

$$\:\underset{5*1}{{L}_{\beta\:}}=[\:{\beta\:}_{1}-{\alpha\:}_{0}{sin}(\delta\:{\phi\:}_{0})\text{}{\beta\:}_{2}-{\alpha\:}_{0}{sin}(\delta\:{\phi\:}_{0})\text{}\cdot\:\cdot\:\cdot\:\text{}{\beta\:}_{5}-{\alpha\:}_{0}{sin}(\delta\:{\phi\:}_{0})\:{]}^{T}$$

(35)

$$\:\underset{5*1}{{L}_{l}}=[\:{l}_{1}-{\alpha\:}_{0}{\varDelta\:}_{0}{cos}(\delta\:{\phi\:}_{0})\text{}{l}_{2}-{\alpha\:}_{0}{\varDelta\:}_{0}{cos}(\delta\:{\phi\:}_{0})\text{}\cdot\:\cdot\:\cdot\:\text{}{l}_{5}-{\alpha\:}_{0}{\varDelta\:}_{0}{cos}(\delta\:{\phi\:}_{0})\:{]}^{T}$$

(36)

where $\:{\beta\:}_{t}$ and $\:{l}_{t}$ represent TFCPC and TFPC multipath errors, and $\:{B}_{\beta\:}$ and $\:{B}_{l}$ are their coefficient matrices, respectively. $\:V$ denotes correction values of the multipath error sequence and t, the subscript, is the serial number of the multipath errors (t = 1, 2, …, 5). The derivation formulas for the correction values of the parameters $\:{x}_{\beta\:}$ and $\:{x}_{l}$ must be estimated as follows,

$$\:\underset{m*m}{N}={\underset{5*m}{B}}^{T}\underset{5*m}{B}$$

(37)

$$\:\underset{m*1}{W}={\underset{m*5}{B}}^{T}\underset{5*1}{L}$$

(38)

$$\:\underset{m*1}{x}={\underset{m*m}{N}}^{-1}\underset{m*1}{W}$$

(39)

where the rank of $\:N$ is equal to the rank of $\:B$, both being 1. Therefore, $\:N$ is a rank-deficient matrix, which leads to non-unique solutions for Eq. (39). To obtain a unique solution, the inverse matrix of $\:N$, denoted as $\:{N}^{-1}$, must be replaced with the pseudoinverse matrix $\:{N}^{-}$, and Eq. (39) should be modified accordingly,

$$\:\underset{m*1}{x}={\underset{m*m}{N}}^{-}\underset{m*1}{W}$$

(40)

The adjusted values of the delay phase, AAF, and path difference are described as follows,

$$\:\left[\begin{array}{c}\stackrel{\varLambda\:}{\alpha\:}\\\:\stackrel{\varLambda\:}{\phi\:}\end{array}\right]=\left[\begin{array}{c}{\alpha\:}_{0}+{x}_{\alpha\:}\\\:{\phi\:}_{0}+{x}_{\phi\:}\end{array}\right]$$

(41)

$$\:\left[\begin{array}{c}\stackrel{\varLambda\:}{\alpha\:}\\\:\stackrel{\varLambda\:}{\phi\:}\\\:\stackrel{\varLambda\:}{\varDelta\:}\end{array}\right]=\left[\begin{array}{c}{\alpha\:}_{0}+{x}_{\alpha\:}\\\:{\phi\:}_{0}+{x}_{\phi\:}\\\:{\varDelta\:}_{0}+{x}_{\varDelta\:}\end{array}\right]$$

(42)

Source link