# 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
#### 3.1.3 Evaluation Indicators for Target Positioning Algorithms  
For the data acquired by simulation analysis, an evaluation index needs to be selected to evaluate the effectiveness of the algorithm. When performing multiple simulation experiments on the target, the obtained positioning result will be randomly distributed near the real position of the target due to the influence of systematic errors and random errors. The distance between the positioning result and the target position reference value is the positioning error. Considering multiple positioning errors as a number array, the evaluation index of positioning accuracy can be developed by data processing.  
There are three commonly used positioning accuracy evaluation indicators:  
(1) Average positioning error \(E_{A}\), which refers to the average value of all positioning errors in a simulation experiment, can be calculated by Equation (35).  
\[E_{A}=\frac{\sum\limits_{i=1}^{n}\Delta y_{n}}{n} \tag{35}\]  
(2) Positioning standard deviation \(E_{STD}\), which refers to the standard deviation of the positioning error array \((\Delta y_{1},\Delta y_{2},\ldots,\Delta y_{n})\), as shown in Equation (36).  
\[E_{STD}=\sqrt{\frac{\sum\limits_{i=1}^{n}\left(y_{i}-\overline{y}\right)}{n-1}} \tag{36}\]  
(3) The circle probability error \(E_{CEP50}\), refers to the radius of the circle centered on the target's true geographic location and containing 50% of the positioning results. Its general form is shown in Equation (37) [88, 89, 90].  
\[\frac{1}{2\pi\sigma_{x}\sigma_{y}\sqrt{1-\rho^{2}}}\iint\limits_{x^{2}+y^{2} \leq R^{2}}\exp\left\{-\frac{1}{2(1-\rho^{2})}\left[\frac{(x-\mu_{x})^{2}}{\sigma _{x}^{2}}-\frac{2\rho(x-\mu_{x})(y-\mu_{y})}{\sigma_{x}\sigma_{y}}+\frac{(y-\mu _{y})^{2}}{\sigma_{y}^{2}}\right]\right\}d_{x}d_{y}=0.5. \tag{37}\]  
The radius value, which is the circle probability error, can be calculated from formula 37. Considering the calculation of Equation (37) is too complicated, another simplified calculation method is given in Equations (38) and (39), which decomposes the positioning result data into \(X\)-axis and \(Y\)-axis components.  
\[\sigma_{x}=\sqrt{\frac{\sum\limits_{i=1}^{n}{(x_{i}-\mu_{x})}}{n-1}},\sigma_{ y}=\sqrt{\frac{\sum\limits_{i=1}^{n}{(y_{i}-\mu_{y})}}{n-1}}. \tag{38}\]  
In the positioning algorithm, the results of the \(X\)-axis and \(Y\)-axis components are both independent, which have different standard deviations due to random errors. Equation (39), which illustrates the quantum relationship between circular probability error and standard deviation, can be derived from Equation (37) through polar coordinates transformation and probability density formula.  
\[E_{CEP}=0.589(\sigma_{x}+\sigma_{y}). \tag{39}\]  
The accuracy of the positioning algorithm has a negative correlation to the values of the three evaluation indicators. For different situations, it is significant to select appropriate indicators to analyze the positioning algorithm. In general, the circular probability error \(E_{CEP50}\) is more in line with the actual positioning error, which includes more than half the positioning result. Therefore, \(E_{CEP50}\) is widely used as an evaluation index in the positioning error analysis process. Unless otherwise specified, the position errors mentioned below all refer to \(E_{CEP50}\).