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Program gama-local
lists separately review of coordinates of fixed and
adjusted points; adjusted constrained coordinates are marked with
*
; see equation (3). Adjusted coordinate standard deviations
m_x and m_y, and values for computing confidence intervals
are given in the listing of adjusted coordinates (Parameters of statistical analysis). In the review index i is the index of unknown
x_i from the system of project equations (1) corresponding to the
point coordinates x and y.
Fixed points ************ point x y ======================================== 1 1054980.484 644498.590 2 1054933.801 643654.101 Adjusted coordinates ******************** i point approximate correction adjusted std.dev conf.i. ====================== value ====== [m] ====== value ========== [mm] === 422 2 x 1055167.22747 -0.00510 1055167.22237 2.7 5.4 3 y 644041.46119 0.00023 644041.46142 2.5 5.1 424 4 X * 1055205.41198 -0.00056 1055205.41142 3.1 6.3 5 Y * 644318.24425 -0.00125 644318.24300 3.6 7.2 |
For adjusted points, program summarizes information on standard ellipses, confidence ellipses, mean square positional errors (m_p), mean coordinate errors (m_xy) and coefficients g characterizing position of approximate coordinates with regard to the confidence ellipse.
Mean errors and parameters of error ellipses ******************************************** point mp mxy mean error ellipse conf.err. ellipse g ========== [mm] == [mm] ==== a [mm] b alpha[g] ==== a' [mm] b' ======== 422 3.6 2.6 2.7 2.5 187.0 6.8 6.4 0.8 424 4.7 3.4 3.7 2.9 131.8 9.5 7.4 0.2 403 5.7 4.0 4.3 3.6 78.9 11.0 9.3 1.1 |
Mean square positional error m_p and mean coordinate error (m_xy) are computed as
m_p = sqrt(m_y^2 + m_x^2), m_xy = m_p / sqrt(2), |
where m_y^2 and m_x^2 are squares of standard deviations (variances) of adjusted points coordinates.
Semimajor and semiminor axes of standard ellipse are denoted as a and b in the listing, bearing of semimajor axis is denoted as alpha and they are computed from covariances of adjusted coordinates
a = sqrt(1/2(cov_yy + cov_xx + c), b = sqrt(1/2(cov_yy + cov_xx - c), c = sqrt( (cov_xx - cov_yy)^2 + 4(cov_xy)^2 ), tan 2alpha = 2(cov_xy) / (cov_xx - cov_yy). |
The angle alpha (the bearing of semimajor axis) is measured clockwise from X axis.
Probability that standard ellipse covers real position of a point is
relatively low. For this reason program gama-local
computes extra
confidence ellipse for which the probability of covering real
point position is equal to the given confidence probability. Both
ellipsy are located in the same center, they share the same bearing of
semimajor axes and they are similar. For lengths of their semi-axis
holds
a' = k_p a, b' = k_p b, |
where k_p is a coefficient computed for the given probability P as defined in Parameters of statistical analysis.
Position of approximate coordinates of an adjusted point with respect to its confidence ellipse are expressed by a coeeficient g Three cases are possible
The coefficient g is calculated from formula
g = sqrt( (a_0 / a')^2 + (b_0/b')^2 ) |
where
b_0 = delta_y cos(alpha) - delta_x sin(alpha), a_0 = delta_y sin(alpha) - delta_x cos(alpha) |
symbol delta is used for correction of approximate coordinates and alpha is bearing of confidence ellipse semimajor axis.
If network contains sets of observed directions, program writes information on corresponding adjusted orientations, standard deviations and confidence intervals. Index i is the same as in the case of adjusted coordinates the index of i-th adjusted unknown in the project equations.
Adjusted bearings ***************** i standpoint approximate correction adjusted std.dev conf.i. ==================== value [g] ==== [g] === value [g] ======= [cc] === 1 1 296.484371 -0.000917 296.483454 5.1 10.3 10 2 96.484371 0.000708 96.485079 5.1 10.4 21 403 20.850571 -0.001953 20.848618 8.8 17.7 |
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