Optimization and Analysis of GPS Elevation Anomaly Models Hua Xisheng, Lei Weigang, Yue Dongjie (School of Civil Engineering, Hohai University, Nanjing 098) rationally studied the significance of data criterion for fitting model parameters, and established elevation anomaly model. The accuracy of the elevation of other points in the GPS level has been significantly improved, and it has good application value.

0 INTRODUCTION A large-scale control network can obtain three-dimensional coordinates of control points using GPS precise positioning. Because the earth elevation must be converted from the established elevation anomaly model to normal height, it can meet the needs of various engineering constructions with daily leveling as a means. The accuracy of GPS leveling depends not only on the accuracy of GPS positioning and the accuracy of leveling measurement, but also on the accuracy of the height anomaly fitting model. In addition, further improving the accuracy of the fitting model has an important role in estimating the reliability of the elevation accuracy of other control points.

For a large-scale control network, the accuracy of the GPS elevation at each point is different after adjustment due to observation errors and graphic effects.

Due to the large-scale high-precision leveling measurement, the workload is large and the time required is long. Therefore, select an appropriate number of GPS control points and measure the precision level, establish an elevation abnormal model in the control area, and then calculate other unmeasured level elevation points Normally high, this will greatly reduce the workload of the field and save time in the field.

In the establishment of the abnormal elevation model, the control point with gross error should be identified and removed from the model to avoid undesirable distortion and distortion of the model. It is necessary to determine the significance of the measured value of each control point on the built model, and the points with significant effects must be included in the built model, so as not to discard it arbitrarily because of too much known data, affecting the accuracy of the model. To further improve the accuracy of the built model, the diagnosis of model data is extremely important.

1. The model of elevation abnormality 1. The secondary model is within a certain range, if the change of normal gravity can be ignored without timing, relative to the reference point P 0, the model of elevation abnormality in this area can be expressed as T is the reference point The height anomaly T, h, is the reference point at x, respectively, and the vertical deviation T in the y direction is the rate of change of the vertical deviation Δx is the coordinate difference between each point and the point P.

Equation (2) is a quadratic surface model with an abnormal elevation at P as a reference point. In a small area, and the area where the anomaly changes in elevation is very gentle, that is, when h 1 km, the quadratic term in equation (2) is not greater than 1 mm. Therefore, in precision GPS leveling, it is generally not appropriate to use a planar model Instead of the quadratic model.

In addition, from the point of view of the quadratic model, if point P is selected far away from the measurement area, it is possible to ignore the abnormal value of the vertical deviation from each point to different parts along the reference point, reducing the accuracy of the model. In addition, the values ​​of Δx and Δy differ greatly, which is also disadvantageous for the estimation of unknown parameters. Therefore, the choice of point P may affect the accuracy of the model.

1.2 The usual engineering control of the quadratic difference model is limited to a small range, and the emphasis is on the relative accuracy and the quality of the results within the measurement area. Therefore, when the elevation anomaly model is established, a point A 1 located near the center of the average elevation anomaly in the measurement area can be selected as a relative reference point.

Then it is easy to list the quadratic model of the abnormal difference in elevation of any point i relative to the reference point A in the measurement area as the vertical deviation in the x and y directions from T to i in the formula T ′ is the rate of change of the vertical deviation Δx Is the coordinate increment from point i to point A.

Models (3) and (4) are based on point A in the middle of the measurement area, and the distance to each control point is relatively uniform. The values ​​of Δx and Δy are small, which intuitively reflects the situation in the measurement area and improves the accuracy of fitting Is beneficial.

2 Model data diagnosis Select a limited number of points to fit a high-quality elevation anomaly model. You must ensure that points that contribute significantly to the fitted model cannot be missed, and points that contain gross errors cannot be included to optimize the model. To achieve the effect of reducing field workload and improving the use of models.

2.1 The principle of data diagnosis Let the expression of the abnormal height fitting model be such that if the i-th group of data is excluded from the n-group data of the fitted model, then the parameter A fitted from the n-group of data will be removed before removing the i-th group of data After the data is A (i), the effect on the parameters is the test distance. In the above formula, the larger D (M, C) is, the greater the distance A moves after excluding the i-th data. That is, D (M, C) reflects the influence of the i-th set of data on A.

Taking the confidence level T, the critical value of D can be inferred that the set of data has a significant impact on the model built.

2. 2 Testing of abnormal measured values ​​Fit the abnormal height model with limited control points. In order to ensure the accuracy and quality of the model, the points containing abnormal observation values ​​should be eliminated.

In order to avoid the data containing gross errors into the modeling and avoid the distortion of the model.

When fitting the height anomaly model, in order to reduce the workload of leveling measurement significantly and take advantage of GPS leveling, it is advisable to use fewer points to participate in the model fitting. The level elevation can be calculated by fitting the model to other points. Generally speaking, there are few points with abnormal observation values. If the variance e is unknown, it is a more rigorous method to use the U test to find the gross difference point with the standardized residual r. .

Taking the confidence level T, when UU, it can be considered that the observation value is abnormal, it should be eliminated and rejected to participate in modeling.

3 Case analysis 3.1 Control network and its control network for observing a gentle terrain along the river, with an average side length of about 1 km, as shown in Figure 1. At the time of observation of each control point, the GPS intensity of 90 minutes observation and sampling interval graph intensity factor was used as the research purpose. Each control point was measured at level II, and 5 side lengths were measured by TC in the network. Using the precise ephemeris to implement the individual adjustment of the GPS network, the error in the plane point of the weakest point is not more than ± 3 mm. The side length calculated inversely from the coordinates is compared with the TC plus the measured side length. GPS positioning has reached a higher accuracy.

3.2 Analysis of the fitted model a. Fit all 18 points with the model of formula (2), and solve the parameters A in formula (2)

b. Normalize the number of elevation anomaly corrections v at each point, and apply U test to determine the abnormal observation value.

The points with abnormal observation values ​​are removed, and the remaining control points are used to fit the model according to formula (2), and data diagnosis is performed for each point.

Model the control points with significant effects. From this model, the normal height of other points is calculated and compared with the measured value of level â…¡.

e. Carry out the calculation and analysis of the above steps with the difference model of formula (4).

The final results of the modeling and calculation of the quadratic model and the difference model respectively ~ Table 4. Taking the difference model as an example, in the full-point fitting without data preprocessing, the accuracy of the model is very low due to the effect of gross error , The error in the fitting m = ± 23. 72 mm. After the gross error elimination and the significance test, only 7 points are needed to establish the difference model, and the error in the fitting is significantly improved. Using this model to estimate the elevation of each point, compared with the level II elevation, the medium error completely meets the level II requirement. In this example, the difference model established by using 7 significant points is significantly better than the model established by all 18 points in terms of the convenience of accuracy and the accuracy of estimating the elevation of other points. An elevation anomaly model built from 15 points after excluding 3 gross errors.

Model error / mm in quadratic model difference model fitting? Analysis of monitoring data? Optimization and analysis of GPS elevation anomaly models such as Hua Xisheng, etc. Point number secondary model difference model model point number secondary model model difference model point number secondary model model difference model secondary model difference model Note: Model â‘  It is the model after removing gross outliers. â‘¡The model after removing gross errors and insignificant points.

The gross difference of the model is not significant. The secondary model difference model model point number difference model model point number difference model Note: The gross difference point is not included.

4 Conclusions a. GPS leveling should use a quadratic or difference model to establish the elevation anomaly model, especially the difference model has a significant advantage. The difference between the estimated height of the fitting and the level Ⅱ is small. The median error is only ± 1.27 mm, which can meet the accuracy requirements of level II measurement.

Due to the influence of many factors in the measurement, some of the measured elevation outliers have gross errors. Before modeling, the determination and elimination of abnormal data is an important step to ensure the quality of the model.

c. The data diagnosis of the built model plays an important role in the optimization of the model, the improvement of accuracy and quality. The points with significant modeling effects must be included in the elevation anomaly model and cannot be omitted. It is not necessary to include all known points in the model establishment, so as to avoid complicating the model, inconvenient to use, and failing to meet the requirements of optimization.

The spatial distribution of the points that have a significant effect on the model will vary with the location, graphics, and elevation distribution of the points. In this case, the points that have a significant effect on the model are those distributed around the control area. In order to make the quality of the built model reliable, the accuracy of the observations at these points must be improved.

1 Fang Ren. Statistical distribution and inspection of measurement errors. Shanghai: China measures 2 Yue Jianping. The accuracy and application of engineering GPS leveling. Surveying and Mapping Bulletin, Hua Xisheng, male, professor, doctoral tutor, mainly engaged in research on safety monitoring and safety monitoring.

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