# Review of Target Geo-Location Algorithms for Aerial Remote Sensing Cameras without Control Points
## 3 Error Analysis Model and Influencing Factors of Positioning Algorithm
### Simulation Analysis Method and Evaluation Index of Positioning Algorithm
#### 3.1.1 Total Differential Method  
Total differentiation is a method of researching the state of a system when the state of the independent variables of a multivariate function changes [74; 75]. Because the calculation process of target positioning is indirectly calculating the target geographical position through various parameters, it can usually be described by a multivariate function composed of multiple elementary functions, which can be expressed by Equation (26).  
\[y=f(x_{1},x_{2},\cdots,x_{n}) \tag{26}\]  
In the equation, parameters \(x_{1},x_{2},\ldots,x_{n}\) represent the position, angle, and distance information required by the target positioning algorithm, and \(y\) is the indirect measurement value, which represents the positioning result. For a multivariate function, the increment of the function can be represented by the total differential; the increment of the above function is:  
\[dy=\frac{\partial f}{\partial x_{1}}dx_{1}+\frac{\partial f}{\partial x_{2}} dx_{2}+\cdots+\frac{\partial f}{\partial x_{n}}dx_{n}. \tag{27}\]  
The systematic and random error of each direct measurement value can be expressed as \(\Delta x_{1},\Delta x_{2},\cdots\Delta x_{n}\), and these errors can be regarded as small values to replace the differential component in Equation (27). Therefore, the approximation error \(\Delta y\) of the system can be calculated by Equation (28).  
\[\Delta y=\frac{\partial f}{\partial x_{1}}\Delta x_{1}+\frac{\partial f}{ \partial x_{2}}\Delta x_{2}+\cdots+\frac{\partial f}{\partial x_{n}}\Delta x _{n}. \tag{28}\]  
Obviously, the positioning error of the target consists of longitude, latitude, and altitude. In the ECEF coordinate system, the target coordinate error is equivalent to the target latitude-longitude error during the process of error analysis. According to the mathematical model of the target positioning algorithm, the positioning error components of the \(X\)-axis (\(M_{X}\)) and \(Y\)-axis (\(M_{Y}\)) directions of the target in the ECEF coordinate system can be obtained as shown in Equations (29) and (30).  
\[M_{X}=\sqrt{\big{(}\frac{\partial X}{\partial X_{s}}\delta X_{s}\big{)}^{2}+ \big{(}\frac{\partial X}{\partial Z_{s}}\delta Z_{s}\big{)}^{2}+\big{(}\frac{ \partial X}{\partial Z_{A}}\delta Z_{A}\big{)}^{2}+\big{(}\frac{\partial X}{ \partial\theta}\delta\theta\big{)}^{2}+\big{(}\frac{\partial X}{\partial \gamma}\delta\gamma\big{)}^{2}+\big{(}\frac{\partial X}{\partial\varphi} \delta\varphi\big{)}^{2}+\cdots}} \tag{29}\]  
\[M_{Y}=\sqrt{\big{(}\frac{\partial Y}{\partial Y_{C}}\delta Y_{C}\big{)}^{2}+ \big{(}\frac{\partial Y}{\partial Z_{s}}\delta Z_{s}\big{)}^{2}+\big{(}\frac{ \partial Y}{\partial Z_{A}}\delta Z_{A}\big{)}^{2}+\big{(}\frac{\partial Y}{ \partial\theta}\delta\theta\big{)}^{2}+\big{(}\frac{\partial Y}{\partial \gamma}\delta\gamma\big{)}^{2}+\big{(}\frac{\partial Y}{\partial\varphi} \delta\varphi\big{)}^{2}+\cdots}} \tag{30}\]  
The error in the \(Z\)-axis direction is only related to the altitude, so it can be found as:  
\[M_{Z}=\delta H. \tag{31}\]  
According to the error synthesis theory [16; 74], the positioning error \(M\) of target can be defined as  
\[M=\sqrt{M_{X}^{2}+M_{Y}^{2}+M_{Z}^{2}}. \tag{32}\]  
Although the traditional total differential method can perform error analysis, the calculation method is too complicated, time-consuming, and the accuracy fluctuates far from satisfactory.