*3.1. KF Based on MCC Derivation*

Assuming the joint PDF of random variables *X* and *Y* as *FXY*(*x*,*y*), the correntropy is defined as follows [47]:

$$\begin{array}{lcl}V(X,Y) &= E[\mathcal{G}\_{\sigma}(X-Y)] \\ &= \int \mathcal{G}\_{\sigma}(x-y) dF\_{XY}(x,y) \\ &= \int \exp\left(-\frac{(X-Y)^{2}}{2\sigma^{2}}\right) dF\_{XY}(x,y) \end{array} \tag{9}$$

where *σ* is the KBW, *E* is the expectation operator, and *G*σ is the non-negative Gaussian kernel function. Furthermore, the Taylor expansion of the above equation is:

$$V(X,Y) = \frac{1}{\sqrt{2\pi}\sigma} \sum\_{n=0}^{\infty} \frac{(-1)^n}{2^n n!} E\left[\frac{(X-Y)^{2n}}{\sigma^{2n}}\right] \tag{10}$$

Here, *V* is essentially a correlation function in the local kernel space controlled by *σ*, as it is the weighted sum of all even order moments of (*X*-*Y*). This localization proves meaningful in measuring the similarity between *X* and *Y* [48,49]. Then, the KF based on MCC (MCC-KF) can be established by optimizing the following loss function [50–52]:

$$J\_{\mathbb{C}} = G\_{\sigma}(||\mathfrak{z}\_{k} - H\mathfrak{x}\_{k}||) + G\_{\sigma}(||\mathfrak{x}\_{k} - \Phi\mathfrak{x}\_{k-1}||)\tag{11}$$

where denotes the Euclidean norm. *<sup>J</sup>*<sup>c</sup> is only a function of *<sup>σ</sup>* [25]. Let *<sup>∂</sup>J<sup>C</sup> <sup>∂</sup>x*<sup>ˆ</sup> *<sup>k</sup>* <sup>=</sup> 0; the estimated state can be obtained as follows [53]:

$$\hat{\mathfrak{x}}\_{k} = \Phi \hat{\mathfrak{x}}\_{k-1} + \frac{G\_{\sigma}(||\mathfrak{z}\_{k} - H\hat{\mathfrak{x}}\_{k}||)}{G\_{\sigma}(||\mathfrak{x}\_{k} - F\hat{\mathfrak{x}}\_{k-1}||)} H^{T}(\mathfrak{z}\_{k} - H\hat{\mathfrak{x}}\_{k}) \tag{12}$$

It tells that the MCC will be achieved if *X* = *Y*, as *Gσ* reaches the upper bound and the PDF of the predicted value and the measured value matched to the maximum extent [54]. The further results can be obtained while *x<sup>k</sup>* ≈ *Fx*ˆ *<sup>k</sup>*−1:

$$\hat{\mathfrak{x}}\_{k} = \hat{\mathfrak{x}}\_{k}^{-} + G\_{\sigma}(||\mathfrak{z}\_{k} - \mathbf{H}\mathfrak{x}\_{k}||)H^{T}(\mathfrak{z}\_{k} - \mathbf{H}\mathfrak{x}\_{k})\tag{13}$$

#### *3.2. AMC-KF Derivation*

The KBW of the originally proposed MCC shown in Equation (13) is usually predefined empirically, which results in the compromise between fast learning initially and fast learning near the optimum point. To derive an adaptive KBW, the loss function is further enhanced as follows:

$$\begin{split} \mathcal{J}\_{\mathbb{C}} &= \quad \mathcal{G}\_{\sigma} \left( \left\| \mathbb{z}\_{k} - H \hat{\mathbf{x}}\_{k} \right\|\_{\mathbf{R}\_{k}^{-1}} \right) + \mathcal{G}\_{\sigma} \left( \left\| \hat{\mathbf{x}}\_{k} - \Phi \hat{\mathbf{x}}\_{k-1} \right\|\_{\mathbf{P}\_{k|k-1}^{-1}} \right) \\ &\leq \mathcal{G}\_{\sigma} \left( \left\| \mathbb{z}\_{k} - H \hat{\mathbf{x}}\_{k} \right\|\_{\mathbf{R}\_{k}^{-1}} \right) + \Lambda \\ &= \frac{1}{N\sqrt{2\pi\sigma}} \sum\_{\mathbf{i}=\mathbf{n}-\mathbf{N}+1}^{\mathbf{n}} \exp\left( \frac{-\left\| \mathbb{z}\_{k} - H \hat{\mathbf{x}}\_{k} \right\|\_{\mathbf{R}}^{2}}{2\sigma^{2}} \right) + \Lambda \end{split} \tag{14}$$

where Λ is a constant overbounded by lim *x*ˆ *<sup>k</sup>*−**Φ***x*ˆ *<sup>k</sup>*−1 *<sup>P</sup>*−<sup>1</sup> *k*|*k*−1 →0 *Gσ x*ˆ *<sup>k</sup>* − **Φ***x*ˆ *<sup>k</sup>*−1*P*−<sup>1</sup> *k*|*k*−1 [47,55].

To search for the proper *σ*, the gradient ascent approach is applied by taking a small step *μ* along the positive gradient, then the *n*th iteration can be expressed as *σ*n+1 = *σ*<sup>n</sup> + *μ*∇*J*<sup>c</sup> [47]. The *J*c can be minimized as follows:

$$\begin{array}{ll} \nabla I\_{\mathbb{C}} &= \frac{\partial I\_{\mathbb{C}}}{\partial \tau} = -\frac{1}{N\sqrt{2}\pi r^{2}} \sum\_{i=n-N+1}^{n} \exp\left(-\frac{\|z\_{i} - H\mathbf{b}\_{i}\|\_{R-1}}{2r^{2}}\right) + \frac{1}{N\sqrt{2}\pi\sigma} \sum\_{i=n-N+1}^{n} \left(\frac{\|z\_{i} - H\mathbf{b}\_{i}\|\_{R-1}}{\sigma^{3}}\right) \exp\left(-\frac{\|z\_{i} - H\mathbf{b}\_{i}\|\_{R-1}}{2r^{2}}\right) \\ &= -\frac{1}{\sqrt{2}\pi r^{2}} \exp\left(-\frac{\|z\_{i} - H\mathbf{b}\_{i}\|\_{R-1}}{2r^{2}}\right) + \frac{\|z\_{i} - H\mathbf{b}\_{i}\|\_{R-1}}{\sqrt{2}\pi r^{4}} \exp\left(-\frac{\|z\_{i} - H\mathbf{b}\_{i}\|\_{R-1}}{2r^{2}}\right) \\ &= 0 \end{array}$$

Then, the closed-formed KBW, which can adaptively adjust according to *R*, is expressed as follows:

$$
\sigma = \sqrt{\frac{\|\mathbf{z}\_k - \mathbf{H}\mathbf{\hat{x}}\_k\|\_{\mathbf{R}^{-1}}}{2}} + \sigma' \tag{15}
$$

Through Equations (4), (12), (13) and (15), the proposed AMC-KF is finally obtained. It should be noted that the exponential part of AMC-KF reduces to constant and is no longer correntropy-based if Equation (15) is applied without the small penalty term *σ*'. The penalty term is artificially added and can be determined according to [56].

#### *3.3. Filter Implementation*

The procedure of the proposed nonlinear strategy is summarized as follows: (1) Removing the ambiguities ∇Δ*N* on each frequency by the DF data-aided AR method. A

threshold and moving average operation are applied to ensure stability and reduce noise. (2) The precise ∇Δ*ϕ* without ambiguities is fed into the AMC-KF for multi-GNSS float solution. The adaptive KBW is used for the prediction and update step during the filtering. Both of them help improve the robustness and accuracy of the float solution. (3) To keep consistency with other RTK structures, the least square ambiguity decorrelation adjustment (LAMBDA) is adopted to transfer the float solution to the fixed solution.

To initialize the proposed filter, the variance-covariance matrix is deduced by the least square method (LS) at the initial epoch. The *F* and *Q* can be defined as an identity matrix and a zero matrix without cycle slips. The noise level for non-difference code and carrier measurements are set to 3 m and 3 cm, respectively [1]. The framework of the proposed nonlinear strategy is depicted in Figure 1. As shown in Equation (8), the window width of the moving average and the threshold in the 'DF Data-aid AR Stage' is usually set to 5 epochs and 0.1 cycles, which implies the influence on the first time to fix ambiguity is tiny and controllable.

**Figure 1.** The procedure of the filter implementation.

#### **4. Test and Result**

To validate the proposed filter strategy, short and long baseline tests are conducted and the traditional DD KF (DD-KF) model mentioned in [1] is also used for comparison. The DD-KF is established in Equations (1)–(4) without the 'DF Data-aid AR Stage'. As the noise of original observations is significantly less than those of wide-lane and ionosphere-free combined measurements, thus, the DF Data-aid AR method is not enabled for the short baseline test. The improvements illustrated in the short baseline test are only beneficial from the AMC-KF method. For the long baseline, the DF Data-aid AR stage is enabled to eliminate atmosphere errors; thus, the improvements in the long baseline test are beneficial both from the AMC-KF and the DF Data-aid AR method. All results obtained are based on post-processing performed on an Intel Core i7 2.30 GHz notebook with 16 GB RAM running on Windows 10.

As Figure 2 shows, the dataset is collected from seven Australia CORS stations (BONE, QCLF, ANGS, STNY, NEWH, GSBN, WBEE) on January 1, 2021, and all formed baselines are elaborated in Table 1. The first six baselines range from 19–60 km and are used for the short baseline test. The last two baselines are formed by (BONE, GSBN, WBEE) and used for long baselines. The sample interval and cut-off elevation for all tests are 30 s and 10◦, respectively.

**Figure 2.** Distribution of CORS station BONE, QCLF, ANGS, STNY, NEWH, GSBN, WBEE. (https: //gnss.ga.gov.au/network (accessed on 1 January 2021)).


**Table 1.** Information for different baselines.
