Method and apparatus of single epoch position bound

A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:

• specifying, in advance, a prior probability density P(x) of a state x;
• specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
• quantifying, during the measurements, quality metrics q associated with the measurements;
• specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
• defining, during the measurements, a posterior probability density P(x|z, q, θ);
• estimating, during the measurements, the state x;
• and
• computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
• wherein quantifying the quality metrics q associated with the measurements comprises excluding one or more outliers of the measurements using a random sample consensus (RANSAC) technique.

A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:

• specifying, in advance, a prior probability density P(x) of a state x;
• specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
• quantifying, during the measurements, quality metrics q associated with the measurements;
• specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
• defining, during the measurements, a posterior probability density P(x|z, q, θ);
• estimating, during the measurements, the state x;
• and
• computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
• wherein the non-Gaussian residual error probability density model is a student-t distribution function, and specifying the residual error probability density model ƒ(r|θ, q) further comprises:
• identifying a model for a pseudorange using the student-t distribution function as:
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• where r is the residual error, ν is a degrees of freedom parameter, σ is a scaling parameter, and Γ is the gamma function.

A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:

• specifying, in advance, a prior probability density P(x) of a state x;
• specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
• quantifying, during the measurements, quality metrics q associated with the measurements;
• specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
• defining, during the measurements, a posterior probability density P(x|z, q, θ);
• estimating, during the measurements, the state x;
• and
• computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
• wherein the non-Gaussian residual error probability density model is a student-t distribution function, and specifying the residual error probability density model ƒ(r|θ, q) further comprises:
• identifying a model for a carrier phase using the student-t distribution function as:
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• where r is the residual error, ν is a degrees of freedom parameter, σ is a scaling parameter, w is a weight, and Γ is the gamma function.

A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:

• specifying, in advance, a prior probability density P(x) of a state x;
• specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
• quantifying, during the measurements, quality metrics q associated with the measurements;
• specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
• defining, during the measurements, a posterior probability density P(x|z, q, θ);
• estimating, during the measurements, the state x;
• and
• computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
• wherein integrating the posterior probability density over the state comprises:
• applying Markov chain Monte Carlo (MCMC) numerical integration using a plurality of interacting chains with modified density functions.

A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:

• specifying, in advance, a prior probability density P(x) of a state x;
• specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
• quantifying, during the measurements, quality metrics q associated with the measurements;
• specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
• defining, during the measurements, a posterior probability density P(x|z, q, θ);
• estimating, during the measurements, the state x;
• and
• computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
• wherein integrating the posterior probability density over the state x comprises:
• performing a first round of sampling by drawing a first set of samples from an unbiased distribution of the posterior probability density and estimating a first location of a quantile on the first set of samples;
• performing a second round of sampling by drawing a second set of samples from a first constrained distribution of the posterior probability density that is determined based on the first location of the quantile;
• estimating a second location of the quantile on the second set of samples, wherein the second location of the quantile is used to set a second constrained distribution for a third round of sampling;
• repeating the sampling using a constrained distribution set in previous sampling until nth round of sampling is completed, where n is a natural number greater than 2;
• determining whether samples of the n round samplings are adequate for computing the protection level;
• and
• combining the samples of n rounds sampling with appropriate weights when the samples of n rounds are adequate for computing the protection level.

A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:

• specifying, in advance, a prior probability density P(x) of a state x;
• specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
• quantifying, during the measurements, quality metrics q associated with the measurements;
• specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
• defining, during the measurements, a posterior probability density P(x|z, q, θ);
• estimating, during the measurements, the state x;
• computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x;
• and
• performing bound propagation using phase change error distribution or data obtained by one or more sensors.

An apparatus for global navigation satellite system (GNSS) measurements, the apparatus comprising:

• a receiver configured to receive a GNSS signal and process the received signal for the measurements;
• and
• a processor configured to:
• specify a prior probability density P(x) of a state x;
• specify a system model h(x) that relates the state x to observables z of the measurements;
• quantify quality metrics q associated with the measurements;
• specify a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ using a set of experimental data, where r is the residual error;
• define a posterior probability density P(x|z, q, θ);
• and
• estimate the state x;
• and
• compute a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
• wherein defining the posterior probability density is based on the formula:
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• where i is a natural number.