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Regularized integer least-squares estimation: Tikhonov’s regularization in a weak GNSS model

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Abstract

The strength of the GNSS precise positioning model degrades in cases of a lack of visible satellites, poor satellite geometry or uneliminated atmospheric delays. The least-squares solution to a weak GNSS model may be unreliable due to a large mean squared error (MSE). Recent studies have reported that Tikhonov’s regularization can decrease the solution’s MSE and improve the success rate of integer ambiguity resolution (IAR), as long as the regularization matrix (or parameter) is properly selected. However, there are two aspects that remain unclear: (i) the optimal regularization matrix to minimize the MSE and (ii) the IAR performance of the regularization method. This contribution focuses on these two issues. First, the “optimal” Tikhonov’s regularization matrix is derived conditioned on an assumption of prior information of the ambiguity. Second, the regularized integer least-squares (regularized ILS) method is compared with the integer least-squares (ILS) method in view of lattice theory. Theoretical analysis shows that regularized ILS can increase the upper and lower bounds of the success rate and reduce the upper bound of the LLL reduction complexity and the upper bound of the search complexity. Experimental assessment based on real observed GPS data further demonstrates that regularized ILS (i) alleviates the LLL reduction complexity, (ii) reduces the computational complexity of determinate-region ambiguity search, and (iii) improves the ambiguity fixing success rate.

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Data availability statement

The datasets in this contribution are available from the Hong Kong Geodetic Survey Service, http://www.geodetic.gov.hk/tc/satref/satref.htm.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (nos. 41504029 and 41631072) and the Natural Science Foundation for Distinguished Young Scholars of Hubei Province of China (no. 2019CFA086). I would like to express my appreciation to Prof. Athanasios Dermanis and two other reviewers, Editor-in-Chief Jürgen Kusche, and Associate Editor Mattia Crespi for their helpful suggestions.

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Authors and Affiliations

  1. Department of Navigation, Naval University of Engineering, Wuhan, China

    Zemin Wu & Shaofeng Bian

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  1. Zemin Wu
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  2. Shaofeng Bian
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Contributions

Conceptualization, software, and writing were performed by ZW; methodology and experiment assessment were performed by ZW and SB. All the authors have read and approved the final manuscript.

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Correspondence to Zemin Wu.

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We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Appendix

Appendix

The original GNSS mathematical model can be expressed as a nonlinear vector function as follows:

$$\begin{array}{cc}{\varvec{s}}=\mathrm{f}\left({\varvec{x}}\right)+{\varvec{\varepsilon}},& {{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}\end{array},$$
(57)

where \({\varvec{s}}\in {\mathbb{R}}^{l}\) is the original double-differenced carrier phase and code observables; \({\varvec{x}}={\left[{\mathfrak{a}}^{T}\boldsymbol{ }\boldsymbol{ }{\mathfrak{b}}^{T}\right]}^{T}\) is a vector of unknown parameters, with integer-valued \(\mathfrak{a}\in {\mathbb{Z}}^{n}\) and real-valued \(\mathfrak{b}\in {\mathbb{R}}^{m}\); and \({\varvec{\varepsilon}}\) is a vector of observation errors. \(\mathrm{f}({\varvec{x}})\) is a nonlinear measurement vector function of parameter vector \({\varvec{x}}\). The VC matrix of observations in \({\varvec{s}}\) is \({{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}\), which is regarded as known a priori. The equation can be extended using a Taylor series around an initial parameter vector \({{\varvec{x}}}_{(0)}\) as

$$\mathrm{f}({\varvec{x}})=\mathrm{f}({{\varvec{x}}}_{(0)})+{\varvec{F}}({\varvec{x}}-{{\varvec{x}}}_{(0)})+ \cdots $$
(58)

where \({\varvec{F}}\) is a partial derivatives matrix of \(\mathrm{f}({\varvec{x}})\) with respect to \({\varvec{x}}\) at \({\varvec{x}}={{\varvec{x}}}_{(0)}\)

$${\varvec{F}}={\left.\frac{\partial \mathrm{f}({\varvec{x}})}{\partial {\varvec{x}}}\right|}_{{\varvec{x}}={{\varvec{x}}}_{(0)}}.$$
(59)

If the initial parameters are adequately near the true values, the second and further terms of the Taylor series can be neglected. We can approximate the following linear equation as

$${\varvec{s}}-\mathrm{f}\left({{\varvec{x}}}_{\left(0\right)}\right)={\varvec{F}}\delta {{\varvec{x}}}_{\left(0\right)}+{\varvec{\varepsilon}}.$$
(60)

Then, we can obtain the estimated unknown parameter vector increment \(\delta {\widehat{{\varvec{x}}}}_{(0)}\) by

$$\delta {\widehat{{\varvec{x}}}}_{(0)}=({{\varvec{F}}}^{T}{{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}^{-1}{\varvec{F}}{)}^{-1}{{\varvec{F}}}^{T}{{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}^{-1}\left({\varvec{s}}-\mathrm{f}\left({{\varvec{x}}}_{\left(0\right)}\right)\right).$$
(61)

If the initial parameters \({{\varvec{x}}}_{(0)}\) are not sufficiently near the true values, we can iteratively improve the estimated parameters as

$$ \left\{ {\begin{array}{*{20}c} {\delta \user2{\hat{x}}_{{(i)}} = (\user2{F}^{T} \user2{Q}_{{\user2{ss}}}^{{ - 1}} \user2{F})^{{ - 1}} \user2{F}^{T} \user2{Q}_{{\user2{ss}}}^{{ - 1}} (\user2{s} - {\text{f}}(\user2{\hat{x}}_{{(i)}} ))} \\ {\user2{\hat{x}}_{{(i + 1)}} = \user2{\hat{x}}_{{(i)}} + \left[\kern-0.15em\left[ {\delta \user2{\hat{x}}_{{(i)}} } \right]\kern-0.15em\right]} \\ {\user2{\hat{x}}_{{(i + 1)}}^{\prime } = \user2{\hat{x}}_{{(i)}} + \delta \user2{\hat{x}}_{{(i)}} } \\ \end{array} } \right. $$
(62)

with

(63)

until all the elements in the observed-minus-computed vector \({\varvec{s}}-\mathrm{f}({\widehat{{\varvec{x}}}}_{\left(i+1\right)}^{\boldsymbol{^{\prime}}})\) are within a user-defined post fit threshold

$$\begin{array}{cc}\forall 1\le k<l:& {{\varvec{e}}}_{k}^{T}[{\varvec{s}}-\mathrm{f}\left({\widehat{{\varvec{x}}}}_{\left(i+1\right)}^{\boldsymbol{^{\prime}}}\right)]{[{\varvec{s}}-\mathrm{f}\left({\widehat{{\varvec{x}}}}_{\left(i+1\right)}^{\boldsymbol{^{\prime}}}\right)]}^{T}{{\varvec{e}}}_{k}\le \alpha {{\varvec{e}}}_{k}^{T}{{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}{{\varvec{e}}}_{k}\end{array},$$
(64)

where means rounding to the nearest integer, \(\alpha \) is a user-defined post fit threshold, and \({{\varvec{e}}}_{k}\) is the unit vector with its \(k\)th entry a 1 and other entries 0. Using \({\widehat{{\varvec{x}}}}_{\left(i+1\right)}\) instead of \({\widehat{{\varvec{x}}}}_{\left(i+1\right)}^{\mathrm{^{\prime}}}\) in the iterations guarantees the integer nature of the ambiguity vector in each step. However, we also need to use \({\widehat{{\varvec{x}}}}_{\left(i+1\right)}^{\mathrm{^{\prime}}}\) to judge when the iterations could terminate. This process is a modification of the well-known Gauss–Newton iteration. If (64) is satisfied after \(j\) iterations, the linearized GNSS model is obtained as

$$\begin{array}{cc}\mathrm{E}({\varvec{y}})={\varvec{A}}{\varvec{a}}+{\varvec{B}}{\varvec{b}},& \mathrm{D}\left({\varvec{y}}\right)={{\varvec{Q}}}_{{\varvec{y}}{\varvec{y}}}\end{array}$$
(65)

with

$${\varvec{y}}={\varvec{s}}-\mathrm{f}\left({\widehat{{\varvec{x}}}}_{(j)}\right)$$
$$[{\varvec{A}}\boldsymbol{ }\boldsymbol{ }{\varvec{B}}]={\varvec{F}}$$
$${[{{\varvec{a}}}^{T} {{\varvec{b}}}^{T}]}^{T}=\delta {{\varvec{x}}}_{(j)}$$
$${{\varvec{Q}}}_{{\varvec{y}}{\varvec{y}}}={{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}$$

and the float solution of the parameter vector is

$${[{\widehat{{\varvec{a}}}}^{T} {\widehat{{\varvec{b}}}}^{T}]}^{T}=\delta {\widehat{{\varvec{x}}}}_{(j)}=({{\varvec{F}}}^{T}{{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}^{-1}{\varvec{F}}{)}^{-1}{{\varvec{F}}}^{T}{{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}^{-1}({\varvec{s}}-\mathrm{f}({\widehat{{\varvec{x}}}}_{\left(j\right)})).$$
(66)

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Wu, Z., Bian, S. Regularized integer least-squares estimation: Tikhonov’s regularization in a weak GNSS model. J Geod 96, 22 (2022). https://doi.org/10.1007/s00190-021-01585-7

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