# Multidimensional Assessment of Lake Water Ecosystem Services Using Remote Sensing
## 2 Materials and Methods
### A Multidimensional Assessment Framework  
The multidimensional ecosystem services assessment framework we proposed in 2020 [5] explores ecosystem services from multiple perspectives. It focuses on the service output from the natural ecosystems and analyzes the service efficiency and trend simultaneously. It considers the output per unit area among different ecosystems and the changes over time in one ecosystem, which have been largely ignored in the traditional assessment methods. Ecosystem services are assessed in three dimensions in this framework, including output (P), efficiency (Q), and trend (D). The output is defined as the total amount of ecosystem services provided by natural ecosystems, and it is the total water storage in this research. The ecosystem service efficiency is defined as the proportion of ecosystem services generated per unit area of ecosystem, which is lake surface area in this paper. The trend index indicates the change in ecosystem efficiency, defined as a vector whose direction and magnitude represent the direction and rate of change, respectively. We used the total water storage as the water provisioning output index, the water storage divided by the lake area as the efficiency index, and the water provisioning annual change vector as the trend index in this paper. We constructed a multidimensional assessment framework based on three axes, output (P axis), efficiency (Q axis), and trend (D axis). We defined eight assessment spaces with various properties in the multidimensional assessment framework. Every object has a unique ecosystem services assessment cuboid, which means how many assessment space properties the object shows in the proposed framework. In Figure 8a, objects 'a' and 'd' are in Space 5, 'b' is in Space 3, and 'c' is in Space 2. Apparently, 'a' and 'd' share the properties of Space 5, and 'b' and 'c' share the properties of Space 3 and Space 2, respectively. However, cuboid 'a' is much larger than cuboid 'd', indicating that 'a' has more significant Space 5 properties. Figure 7: An illustration of lake water storage calculation. A z-score describes the position of a raw score in terms of its distance from the mean, which is measured in standard deviation units. The z-score is positive if the value is above the mean, and negative if it is below the mean. It allows the comparison of scores on different kinds of variables by standardizing the distribution. A z-score can be calculated by the following formula:  
\[x^{*}=\frac{x-\overline{x}}{\sigma} \tag{3}\]  
where \(x^{*}\) is the z-score, \(\overline{x}\) is the mean of the raw scores, \(\sigma\) is the standard deviation of the raw scores. Observations are converted to a standard normal distribution via the z-score standardization. The mean of the standardized data is 0, and the standard deviation is 1. At the same time, the position of every object in all observations is revealed. For example, where a z-score is equal to 2, the observation is larger than 97.7% of the overall data. Conversely, when a z-score is equal to \(-2\), the observation is smaller than 97.7% of the overall data (Figure 8b). We defined P-score, Q-score, and D-score (Table 2) according to the statistical significance of z-scores to express the regional relative level of ecosystem services in each county. Figure 8: The multidimensional framework for evaluating ecosystem services. The multidimensional assessment scores (i.e., P, Q, and D) follow a standard normal distribution. In other words, about 68% of the scores fall within one standard deviation away from the mean; about 95% of the scores fall within two standard deviations; about 99.7% of the scores fall within three standard deviations (see Figure 8b). We identified scores with absolute values greater than three as extreme objects. We then converted the multidimensional assessment results to 'Q-D,' P-D,' P-Q' distributions by slicing the multidimensional assessment framework on the P, Q, D plane, respectively (Figure 8c).