## The CRLB for standalone spherical positioning

When LBL positioning is used alone, without other complementary references, the precision of such a solution is based on (1) the precision of the range measurements, (2) the geometry of the fixed transponders and mobile host, (3) the accuracy of estimate of the speed of sound and (4) the uncertainty in the estimated location of the fixed seafloor transponders. We can

consider each of these sources of uncertainty by applying the CRLB framework to the spherical positioning measurement model described in Section 3.2. The range measurements are assembled into an measurement vector of length n.

where ft( ) is the non-linear function for spherical positioning (equation (1)) and wr is a zero mean random vector with covariance Er.

The CRLB is calculated by linearizing the measurement model about an operating point, xh¡>. The result is summarized by the first derivative of the measurement equation evaluated at the operating point, i.e., the Jacobian matrix C. For the linearized measurement model with additive Gaussian noise, the CRLB is a matrix combination of the Jacobian, representing the current system geometry, and the measurement covariance quantifying the observation uncertainty.

The CRLB is the best-case performance of an unbiased estimator designed to estimate the mobile host position based on uncertain range observations. The CRLB matrix is the minimum value of the covariance matrix for any unbiased estimate of position, i.e.,

where S^is the unknown true position of the host and xh is the estimated mobile host position. To summarize, the CRLB is a best-case estimate of the state covariance of the position solution as expressed in based on the geometry of the static acoustic beacons, the location of the host relative to the beacons and the range uncertainty.

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