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The measurement errors consist of two parts: systematic error and random error. For many techniques, the systematic errors have been greatly eliminated or supressed due to careful calibration, investigation and modeling. Therefore the measurement error is considered to be primarily random error and to obey Gaussian distribution. In most cases of data analysis, we have a good apriori model for the parameters hence only the adjustments of the apriori model are estimated. Since the adjustments are usually several orders smaller than the parameters, we can use the linear algebra to estimate the adjustments. For the errors obeying Gaussian distribution, the optimal solution is the least squares estimation.
Assume the linearlized observation equation is: 
           Y1 = A1 X + e1                    (1)
    where Y1 are the observables
          X are the estimated parameters 
          A1 is the design matrix
          e1 is the measurement error with variance s12 
    The normal equation is:
          (A1T W1 A1) X = (A1T W1) Y1        (2)
  where W1 = s1-2
    The least squares solution is:
          X = (A1T W1 A1)-1 (A1T W1) Y1        (3)
    If there is another observation:
               Y2 = A2 X + e2                    (4)
  where the variance of e2 is s22 
    The combined normal equation is:
      (A1T W1 A1 + A2T W2 A2) X = (A1T W1) Y1 + (A2T W2) Y2        (5)
    The combined solution is:
      X = (A1T W1 A1 + A2T W2 A2)-1 [(A1T W1) Y1 + (A2T W2) Y2]        (6)