Weighted Least Squares Fundamentals
The question asks to identify the correct statements regarding Weighted Least Squares (WLS) adjustment.
WLS is a method used in estimation and adjustment problems, particularly when observations have different levels of reliability (variance).
Analysis of Statements
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Statement 1: Weighted sum of the squares of the residuals is minimized.
This is the defining principle of WLS. Unlike Ordinary Least Squares (OLS) which minimizes $ \sum e_i^2 $, WLS minimizes the weighted sum $ \sum w_i e_i^2 $, where $ w_i $ is the weight assigned to the $ i^{th} $ residual $ e_i $. This statement is correct.
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Statement 2: The expected value of the residuals is equal to zero.
A fundamental property of least squares methods, including WLS, is that the estimation process results in residuals whose expected value is zero ($ E(e_i) = 0 $). This indicates that, on average, the estimated model fits the data without systematic bias. This statement is correct.
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Statement 3: Redundancy of observations is maximized.
Redundancy relates to the number of excess observations beyond what is strictly necessary for a unique solution. While WLS utilizes all observations, its primary goal is not to maximize redundancy but to achieve the best estimate based on the weighted observations. Maximizing redundancy is a design consideration, not a direct outcome of the WLS algorithm itself. This statement is incorrect.
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Statement 4: Weights are taken inversely proportional to the variance of the observations.
This is the standard procedure in WLS. Observations assumed to be more precise (having lower variance, $ \sigma_i^2 $) are assigned higher weights ($ w_i $), typically $ w_i = 1/\sigma_i^2 $. Conversely, less precise observations (higher variance) receive lower weights. This ensures that more reliable data contributes more significantly to the final estimate. This statement is correct.
Conclusion
Based on the analysis, the correct statements are:
- Weighted sum of the squares of the residuals is minimized.
- The expected value of the residuals is equal to zero.
- Weights are taken inversely proportional to the variance of the observations.
Therefore, the correct options correspond to statements 1, 2, and 4.
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
