Chi-square Test for Gross Errors in Least Squares
The Chi-square ($\chi^2$) test is applied in least squares adjustment primarily to assess the overall quality of the observational data and the validity of the adjustment model. It serves as a crucial statistical check.
Purpose of the Test:- It tests the hypothesis that the actual variance of the observations is consistent with the assumed variance or variance derived from the adjustment.
- A significant result from the $\chi^2$ test indicates that the observed deviations (residuals) are statistically larger than expected under the assumption of purely random errors.
The most common and critical application is to detect the potential presence of gross errors (outliers or blunders) within the initial set of observations. If the test indicates a poor fit (i.e., the calculated $\chi^2$ value exceeds the critical value), it suggests that the observations likely contain systematic mistakes or significant outliers, rather than just random errors.
Therefore, the Chi-square test helps validate the reliability of the input data before or after the adjustment process.
The correct option is B, which identifies the detection of gross errors in observations as the primary use of the Chi-square test in this context.
Important Questions from Errors in Observations
The carrier phase observation model in GNSS is given as $$ \phi_A^i = f \delta^i - \frac{\rho_A^i}{\lambda} - f \delta_A + N_A^i - f \delta_{\text{iono}} + f \delta_{\text{tropo}} + \epsilon $$ where $\phi_A^i$ is the observed carrier phase in cycles, $f$ is the frequency of the carrier in hertz, and $\lambda$ is the wavelength of the carrier in meters.
What is the unit of the ionospheric ($\delta_{\text{iono}}$) and tropospheric ($\delta_{\text{tropo}}$) delay terms in the given equation?- In GNSS positioning, the cycle slips are the most detrimental for estimating _______.
- According to the first order ionospheric delay term, the time delay experienced by the GNSS signal is directly proportional to the Total Electron Content (TEC) in the ionosphere, and inversely proportional to the square of the frequency of the carrier wave. Based on this, the GPS L2 (1227.60 MHz) carrier is slower than the GPS L1 (1575.42 MHz) carrier by a factor of ________ for a given TEC (Rounded off to the nearest integer).
- In the context of Global Navigation Satellite System positioning, the Saastamoinen model provides a correction for ________.
- In the choke ring antenna there are concentric cylinders placed around the antenna that are of a certain depth to minimize the multipath effect. If the signal wavelength is $\lambda$, then the depth of the cylinders in the choke ring antenna should be
