A method comprising: receiving signals from a plurality of global navigation satellite system (GNSS) satellites at a vehicle; obtaining inertial measurements from one or more inertial sensors on the vehicle; determining a main navigation solution for one or more of roll, pitch, platform heading, and true heading for the vehicle using data from the signals from the plurality of GNSS satellites and the inertial measurements; determining a plurality of sub-solutions for the main navigation solution using solution separation; determining a separation between the main navigation solution and each of the sub-solutions: determining a separation variance between the main navigation solution and each of the sub-solutions; determining a threshold for detection of a satellite failure based on the separation variances and an allowable false alert rate; and determining a protection limit that bounds error in the main navigation solution based on the threshold and an allowable missed detection probability, wherein determining a protection limit further comprises: determining a plurality of limits, one for each sub-solution, by multiplying an estimation error variance for the respective sub-solution by a multiplier set based on the allowable missed detection probability, wherein the estimation error variance for each sub-solution is calculated by multiplying together a transpose of an error mapping vector for roll, pitch, platform heading, or true heading, an error covariance matrix for the sub-solution, and the error mapping vector wherein the error mapping vector maps a portion of the errors in a state vector to its effect on a respective measurement and the estimation error variance is the variance of an estimation error for a respective measurement; and selecting a limit from the plurality of limits having the largest value as the protection limit. receiving signals from a plurality of global navigation satellite system (GNSS) satellites at a vehicle; obtaining inertial measurements from one or more inertial sensors on the vehicle; determining a main navigation solution for one or more of roll, pitch, platform heading, and true heading for the vehicle using data from the signals from the plurality of GNSS satellites and the inertial measurements; determining a plurality of sub-solutions for the main navigation solution using solution separation; determining a separation between the main navigation solution and each of the sub-solutions: determining a separation variance between the main navigation solution and each of the sub-solutions; determining a threshold for detection of a satellite failure based on the separation variances and an allowable false alert rate; and determining a protection limit that bounds error in the main navigation solution based on the threshold and an allowable missed detection probability, wherein determining a protection limit further comprises: determining a plurality of limits, one for each sub-solution, by multiplying an estimation error variance for the respective sub-solution by a multiplier set based on the allowable missed detection probability, wherein the estimation error variance for each sub-solution is calculated by multiplying together a transpose of an error mapping vector for roll, pitch, platform heading, or true heading, an error covariance matrix for the sub-solution, and the error mapping vector wherein the error mapping vector maps a portion of the errors in a state vector to its effect on a respective measurement and the estimation error variance is the variance of an estimation error for a respective measurement; and selecting a limit from the plurality of limits having the largest value as the protection limit. <EOS>