10. Theory of measurement errors in geodesy

10.1. General concepts about measurements

Comparison of any value with another homogeneous value, taken as a unit, is called a measurement , and the resulting numerical value is the result of measurement .

There are measurements of direct (immediate) and indirect. Basic equation of direct measurement

λ = N ∙ K

where λ is the measurement result; K - the value adopted for the unit of measurement (comparison); N   - an abstract number, showing how many times λ is more than N.

Indirect measurements are such measurements that are obtained by the formulas relating the values ​​of measured physical quantities with the values ​​of other physical quantities obtained from direct measurements and which are the arguments of these formulas.

Indirect measurement equation

λ = f (λ1, λ2, λ3, ..., λn).

10.2. Measurement errors

The measurement process takes place in time and certain conditions, it involves a measurement object, a measuring instrument, an observer and the environment in which measurements are made. In this regard, the measurement results are influenced by the quality of the measuring devices, the qualifications of the observer, the state of the object being measured and changes in the medium over time. When multiple measurements of the same value due to the influence of these factors, the measurement results may differ from each other and do not coincide with the value of the measured value. The difference between the measurement result and the actual value of the measured value is called the measurement result error .

The nature and properties of errors are divided into:

• rough;

• systematic ;

• random .

Gross errors or miscalculations are easy to detect with repeated measurements or with careful attention to measurements.

Systematic errors are those that act according to certain laws and retain the same sign. Systematic errors can be taken into account in the measurement results, if we find a functional dependence and use it to eliminate the error or reduce it to a small value.

Random errors are the result of several causes. The magnitude of the random error depends on who measures what method and in what conditions .

These errors are called random because each of the factors acts randomly. They cannot be eliminated , but the influence can be reduced by increasing the number of measurements .

10.3. Properties of random measurement errors

Error theory studies only random errors. By random error, hereinafter, we mean the difference

Δ i = X - ℓi

Where   Δ i - true random error; X is the true value; ℓi is the measured value.

Random errors have the following properties:

1. The smaller the absolute value of the random error, the more often it occurs in measurements.

2. Random errors of the same magnitude are equally common in measurements.

3. Under these measurement conditions, the magnitude of the random error in absolute value does not exceed a certain limit. Under these conditions means the same device, the same observer, the same parameters of the environment. Such measurements are called equal.

4. The arithmetic mean of random errors tends to zero with an unlimited increase in the number of measurements.

The first three properties of random errors are fairly obvious. The fourth property follows from the second.

If Δ1, Δ2, Δ3, ..., Δ n are random errors of individual measurements, where n is the number of measurements, then the fourth property of random errors is mathematically expressed

The limit of this ratio will be equal to zero, because in the numerator the sum of random errors will be a finite value, since the positive and negative random errors during addition will be compensated.

To keep the record compact, Gauss suggested writing the sum with the symbol

,

then

10.4. Evaluation of the accuracy of measurement results

By measurement accuracy is meant the degree of closeness of the measurement result to the true value of the measured quantity. The accuracy of the measurement result depends on the measurement conditions.

For equal results of measurements, the measure of accuracy is the mean square error m , determined by the Gauss formula:

.

The standard error is stable with a small number of measurements.

The marginal error.

Due to the third property, random errors exceeding 2 m in absolute value are rare (5 per 100 measurements). Even less often, errors are greater than 3 m (3 out of 1000 measurements). Therefore, the tripled error is called the marginal error

For extremely accurate measurements, take the marginal error

All of the above errors are called absolute . In geodesy, the relative error is used as special characteristics of measurement accuracy - the ratio of the absolute error to the average value of the measured value, which is expressed as a simple fraction with a unit in the numerator, for example

10.5. Mean square error of the general form function

In most cases, geodetic measurements are performed to determine the values ​​of other quantities associated with the measured functional dependence.

For example:

D = K · n;

h = 3 - P;

h = S · tgν.

To judge the accuracy obtained in this case, it is necessary to determine the mean square error of the function from the mean square errors of the initial values, which in turn may be the results of measurements or functions of the results of measurements.

Let u = f ( X, Y, Z ) be some function of independent values ​​X, Y, Z, measured or calculated with mean square errors mx, my, mz .

Differentiate the function for all variables and get

.

In this formula, infinitesimal increments — differentials — are replaced by true errors. Get the expression

,

where ΔX, ΔY, ΔZ are true errors.

Let's move from true errors to mean square errors. To do this, suppose that X, Y, Z is measured n times, where you can read . According to the number of measurements we make n equalities.

We make each of the equalities in a square, add and divide by n

And since ; etc.,

that

Where are partial derivatives of this function, calculated for the corresponding argument values.

10.6. Mathematical processing of the results of equal measurements

Arithmetic average (arithmetic mean). If there are a number of results of equal measurements ℓ1, 2, ..., ℓ n of the same value X, then there is no reason to give preference to any of these values. In this case, the final value X is taken as the value calculated as the arithmetic average of all the results:

Random errors are received as

Adding the left and right sides of these equalities, we get

From here;

,

Based on the fourth property at

Therefore, with a large number of measurements, the arithmetic average is equal to the true value of X. This allows the arithmetic average to be used as the final result of the measurements made. Otherwise it is called the most likely value of the measured value .

Monitoring the calculation of the arithmetic mean is carried out by the most probable errors δ.

Adding equations get .

This property of the most probable errors allows to control the correctness of the calculation of the arithmetic mean.

Mean square error of the arithmetic mean. To calculate the mean square error M of the arithmetic mean, use the formula

from which it follows that the mean square error of the arithmetic mean in times less than the standard error of a single measurement.

Mean square errors expressed through the most probable errors.

Using the deviations (the most probable errors), the mean square error of the deviation m of one measurement is calculated using the Bassel formula

The standard deviation M of the arithmetic center in this case is calculated by the formula

10.7. Equivalent measurements. Concept of weight measurement. Formula of the general arithmetic mean or weight average

If the measurements were not performed under the same conditions, the results cannot be considered equally reliable. Such measurements are called non-equal. For example, the same angle can be measured with an accurate and technical theodolite. The results of these measurements will be unequal.

A measure of comparing the results for non-equivalent measurements, i.e. The measure of the relative value of the obtained unequal results is the weight of the measurement result.

Weight expresses, as it were, the degree of trust that is being shown to this result in comparison with other results.

The more reliable the result, the greater its weight. Weight is defined as the reciprocal of the square of the mean square error

If, for example, there are two unequal-length values ​​of the length of the line 220.35 ± 0.1 m, 220.35 ± 0.2 m, then the numbers P1 and P2 can be taken as weights:

Weights can be multiplied or divided, but by the same number. Dividing the weights calculated in the example by 25, we get p1 = 4 and p2 = 1.

Since p1> p2, the first measurement is more accurate.

Suppose there are a number of equally accurate measurement results. , for which the mean square error m is calculated, the arithmetic average of the series of measurements and the mean square error of M. Based on the determination of weight, the weight p of an individual measurement and the weight of the arithmetic mean P will be

Multiplying the weights by m 2, they have P = 1, P = n, therefore, the weight of the arithmetic mean is greater than the weight of an individual measurement n times, n is the number of measurements from which this arithmetic mean is calculated.

Otherwise, the weight of the measurement result is the number of equal measurements, from which this unequal measurement result is obtained as an arithmetic average.

Consider the derivation of the formula for the general arithmetic mean or weight average .

Let the value has a number of equal measurements:

Р1, Р2 ..... Рk, - not the same number of measurements. Since the measurements are equal, to obtain the most probable value, it is necessary to form the arithmetic average of all the measurement results

Having broken now the considered series of equal measurements into k groups, we shall form the arithmetic means over the groups L ', L' '... L (k). The resulting arithmetic averages can be considered as new results of measurements of the same magnitude, but already unequal. Thus, instead of the original series of equal measurements for a certain quantity, we obtained a new series of non-equivalent measurements L ', L' '..... L (k), with weights Р1, Р2 ..... Рк. According to non-uniform measurements, the arithmetic mean ℓp is determined by the formula

The resulting value is called the total arithmetic mean or weight average.

The total arithmetic average of the data of unequal-current measurements is equal to the sum of the products of each measurement by its weight, divided by the sum of the weights. It is the most likely value of the measured value.

In the same way as in ravnotochnyh measurements, to estimate the accuracy of a separate result and arithmetic mean, when estimating ratios measurements, determine the standard error of a unit of weight

and mean square error of weight mean

Where - evasion of individual measurement results from the general arithmetic mean. To control the correctness of calculations, the property is used.

To control the correctness of calculations, the property is used.

.

10.8. Questions for self-control

1. What is called measurement?

2. What are gross, systematic and random errors?

3. Which measurements are called equipotential and which are non-equilibrium?

4. What are the basic properties of random measurement errors?

5. How to determine the most probable value of the measured value with equal and non-linear measurements?

6. What are the limit, absolute and relative errors?

7. How to determine the mean square error of a function of a general form?

8. What is weight measurement?