Survey Adjustments and Theory of Errors
Contents
Recall the Principle of Least Squares 3
Describe the Laws of Weights 4
Recall the Direct Observations of Equal Weight on a Single Unknown Quantity 5
Recall the Concept of Direct Observations of Unequal Weight on a Single Unknown Quantity 5
Describe the Methods of Determination of Most Probable Values 7
Recall the Adjustments of Various Figures: i. Two Connected Triangles ii. Quadrilateral Triangle 9
Define the Following Terms
i. Independent and Conditioned Quality:
Independent quality refers to a characteristic or property of an object or system that can be measured or evaluated without considering any other factors. It is a standalone characteristic and is not affected by other properties of the object or system.
Conditioned quality refers to a characteristic or property of an object or system that is dependent on or conditioned by other factors. It is a secondary characteristic that is affected by the values of other properties or characteristics of the object or system.
ii. Direct and Indirect Observations:
Direct observations are measurements or evaluations that are made directly on the object or system being studied. The measurement or evaluation is performed using an instrument or device, such as a ruler or thermometer.
Indirect observations are measurements or evaluations that are made based on the results of other measurements or evaluations. The measurement or evaluation is performed by making inferences or calculations based on the results of other observations.
iii. Weight of Observations:
Weight of observations refers to the importance or significance given to a particular observation or measurement. It is used to determine the relative importance of different observations in a particular study or experiment. The weight of an observation can be determined based on factors such as accuracy, reliability, and consistency of the measurement.
iv. Observed Value of a Quality:
Observed value of a quality refers to the value obtained from a measurement or evaluation of a particular characteristic or property of an object or system. It is the result of the direct or indirect observation of the quality.
v. True Value of a Quality:
True value of a quality refers to the actual value of a particular characteristic or property of an object or system. It is the value that would be obtained if an unlimited number of precise and accurate measurements were made.
vi. Most Probable Value:
Most probable value refers to the value of a characteristic or property of an object or system that is most likely to be true based on the results of a particular study or experiment. It is calculated using statistical methods and is based on the observed values of the quality.
vii. True and Most Probable Errors:
True error refers to the difference between the true value of a characteristic or property of an object or system and the observed value of that quality. It is a measure of the accuracy of the observation or measurement.
Most probable error refers to the difference between the most probable value of a characteristic or property of an object or system and the observed value of that quality. It is a measure of the uncertainty or variability of the observation or measurement.
viii. Observation Equation:
Observation equation refers to the mathematical expression used to represent the relationship between the observed value of a quality and its true value. It takes into account the errors and uncertainties in the observation or measurement and is used to calculate the most probable value of the quality. The observation equation is commonly expressed as:
x = X + e
Where:
x = observed value of the quality
X = true value of the quality
e = error or uncertainty in the observation or measurement
Recall the Principle of Least Squares
The principle of least squares is a mathematical method used to find the best approximation of the relationship between two or more variables in a set of data. It is used to find the line of best fit or regression line that best represents the relationship between the variables. The principle of least squares is based on the idea that the sum of the squares of the differences between the observed values and the predicted values should be minimised.
In mathematical terms, the principle of least squares states that the best approximation of the regression line can be obtained by minimising the sum of the squares of the differences between the observed values and the predicted values. This is expressed as:
Σ(y – ŷ)2
Where:
y = observed value
ŷ = predicted value
The principle of least squares can be used to find the regression line for simple linear regression (one independent variable) and multiple linear regression (more than one independent variable). In both cases, the regression line is found by minimising the sum of the squares of the differences between the observed values and the predicted values.
The principle of least squares is widely used in many fields, including statistics, engineering, and economics, to analyze and interpret data. It provides a simple and effective way to estimate the relationship between variables and can be used to make predictions about future values based on the data.
Describe the Laws of Weights
The laws of weights are used to determine the weight of each observation in a set of data when analyzing and interpreting the relationship between variables. The weight of an observation reflects the reliability and precision of that particular observation and helps to determine the most accurate estimate of the true value. There are several laws of weights that are used to assign weights to observations, including:
- The law of inverse variance: The weight of an observation is proportional to the reciprocal of the variance or standard deviation of that observation. This law assumes that observations with smaller variance are more reliable and precise than observations with larger variance.
- The law of equal weight: All observations are given the same weight, regardless of their variance or precision. This law is typically used when there is no information available about the variance of the observations.
- The law of proportional weight: The weight of an observation is proportional to some function of the variance of that observation. This law is used when there is some information available about the variance of the observations, but not enough information to determine the exact variance of each observation.
The choice of which law of weights to use will depend on the nature of the data and the research question being investigated. In general, the law of inverse variance is the most commonly used law of weights, as it provides the most accurate estimate of the true value.
It is important to note that the laws of weights are used to determine the weight of each observation in the analysis and interpretation of data, not to determine the true value of a quality. The true value of a quality can only be determined by making direct observations or by using indirect observations and applying the laws of weights.
Recall the Direct Observations of Equal Weight on a Single Unknown Quantity
A direct observation of equal weight on a single unknown quantity is a method of determining the true value of a quantity based on a set of equally weighted observations. In this method, each observation is given equal weight, regardless of its reliability or precision, and the true value of the quantity is calculated as the mean of the observations.
For example, consider a scenario where the true value of a quantity is unknown and 10 observations are made. Each observation is given an equal weight, and the mean of the 10 observations is calculated. This mean is considered the most probable value of the true value of the quantity, as it takes into account all 10 observations.
The direct observation of equal weight method is simple and straightforward, and is often used when there is limited information available about the reliability or precision of the observations. However, it can lead to an inaccurate estimate of the true value if some of the observations are unreliable or imprecise. In such cases, it may be more appropriate to use the law of inverse variance or another law of weights to determine the weight of each observation.
It is important to note that the direct observation of equal weight method is only applicable to single unknown quantities, where only one quantity is being measured. For multiple unknown quantities, other methods, such as multiple regression analysis, may be more appropriate.
Recall the Concept of Direct Observations of Unequal Weight on a Single Unknown Quantity
A direct observation of unequal weight on a single unknown quantity is a method of determining the true value of a quantity based on a set of observations that are given different weights. In this method, each observation is given a weight that reflects its reliability and precision, and the true value of the quantity is calculated as the weighted mean of the observations.
For example, consider a scenario where the true value of a quantity is unknown and 10 observations are made. Some observations are considered more reliable or precise than others, and are given a higher weight. The weighted mean of the 10 observations is calculated, taking into account the different weights of each observation. This weighted mean is considered the most probable value of the true value of the quantity, as it takes into account both the observations and their relative reliability and precision.
The direct observation of unequal weight method provides a more accurate estimate of the true value of a quantity than the direct observation of equal weight method, as it accounts for the varying reliability and precision of the observations. The weights assigned to each observation can be determined using the law of inverse variance or another law of weights.
It is important to note that the direct observation of unequal weight method is only applicable to single unknown quantities, where only one quantity is being measured. For multiple unknown quantities, other methods, such as multiple regression analysis, may be more appropriate.
Define Normal Equation
The normal equation is a mathematical formula used in linear regression to determine the coefficients of the regression line that best fits a set of data points. The normal equation is a closed-form solution to the linear regression problem and provides the exact solution to the coefficients without any iteration.
The normal equation is derived from the principle of least squares, which states that the line of best fit is the line that minimises the sum of the squared residuals between the observed values and the values predicted by the regression line. The normal equation calculates the coefficients of the regression line by solving the matrix equation that represents the principle of least squares.
The normal equation is given by the formula:
θ = (XT X)-1XT y
where θ is a vector of the coefficients of the regression line, X is a matrix of the independent variables, y is a vector of the dependent variables, and T represents the transpose of a matrix.
The normal equation provides a fast and efficient solution to linear regression, but has some limitations. For example, it requires the inversion of a large matrix, which can be computationally expensive when dealing with large datasets. Additionally, the normal equation is sensitive to singular matrices and may not provide a solution when there are multiple independent variables with high correlations. In such cases, other methods, such as gradient descent, may be more appropriate.
Describe the Methods of Determination of Most Probable Values
The most probable value (MPV) is the value of a quantity that is most likely to be the true value based on a set of observations. There are several methods for determining the most probable value of a quantity, including:
- The method of equal weight: This method involves calculating the average of all observations, giving equal weight to each observation. The average is considered the most probable value of the quantity.
- The method of inverse weight: This method involves weighting each observation by its precision, where more precise observations are given more weight. The weighted average of the observations is considered the most probable value of the quantity.
- The method of maximum likelihood: This method involves finding the value of the quantity that maximises the likelihood function, which represents the probability of obtaining the observed data given the value of the quantity. The value that maximises the likelihood function is considered the most probable value of the quantity.
- The method of Bayesian estimation: This method involves using Bayes’ theorem to update the prior probability distribution of the quantity based on the observed data. The most probable value of the quantity is calculated as the mean of the updated probability distribution.
The most appropriate method for determining the most probable value of a quantity will depend on the nature of the observations, the precision and reliability of the data, and the underlying assumptions about the distribution of the quantity. Each of these methods provides a different estimate of the most probable value, and the choice of method will have an impact on the accuracy of the estimate.
Recall the Following Types of Triangular Adjustments: i. Station Adjustment ii. Angle Adjustment iii. Figure Adjustment
Triangular adjustments are a type of mathematical method used to determine the most probable values of the unknowns in a network of triangles in surveying and geodesy. There are three main types of triangular adjustments:
- Station adjustment: In this type of adjustment, the most probable values of the coordinates of the survey stations are determined. The observations used in this adjustment are usually horizontal or vertical angles and distances measured between the stations. The goal of the adjustment is to determine the most probable values of the station coordinates that are consistent with the observations.
- Angle adjustment: In this type of adjustment, the most probable values of the angles between the lines connecting the survey stations are determined. The observations used in this adjustment are usually horizontal or vertical angles and distances measured between the stations. The goal of the adjustment is to determine the most probable values of the angles that are consistent with the observations.
- Figure adjustment: In this type of adjustment, the most probable values of the geodetic parameters of a network of triangles, such as the scale factor and the orientation, are determined. The observations used in this adjustment are usually horizontal or vertical angles and distances measured between the stations. The goal of the adjustment is to determine the most probable values of the geodetic parameters that are consistent with the observations.
Each of these adjustments has its own specific mathematical techniques and algorithms, and the choice of adjustment will depend on the nature of the observations and the goals of the survey. The results of the adjustment can be used to improve the accuracy of the survey measurements and to determine the most probable values of the unknowns in the network.
Recall the Adjustments of Various Figures: i. Two Connected Triangles ii. Quadrilateral Triangle
The adjustments of various figures are methods used to determine the most probable values of the unknowns in a network of triangles or quadrilaterals in surveying and geodesy. There are two main types of adjustments:
- Two Connected Triangles: In this type of adjustment, the most probable values of the unknowns in two connected triangles are determined. The observations used in this adjustment are usually horizontal or vertical angles and distances measured between the stations. The goal of the adjustment is to determine the most probable values of the unknowns in the triangles that are consistent with the observations.
- Quadrilateral Triangle: In this type of adjustment, the most probable values of the unknowns in a network of quadrilaterals and triangles are determined. The observations used in this adjustment are usually horizontal or vertical angles and distances measured between the stations. The goal of the adjustment is to determine the most probable values of the unknowns in the quadrilaterals and triangles that are consistent with the observations.
Each of these adjustments has its own specific mathematical techniques and algorithms, and the choice of adjustment will depend on the nature of the observations and the goals of the survey. The results of the adjustment can be used to improve the accuracy of the survey measurements and to determine the most probable values of the unknowns in the network.