The standard form of the logistic function is:

In this equation, L is the upper limit (the maximum possible turbidity), and β and α are parameters to be determined. The graph of this function is S-shaped: when x is small, y increases slowly; as x increases, y rises rapidly; it then gradually levels off and approaches L ([2]).

By transforming Equation (1), we obtain:

Equation (2) is called the Logit transformation, written as z = ln ( y / ( L − y ) ) . After this transformation, z and x have a linear relationship. Therefore, we can first compute Z from the original turbidity values y, then apply linear regression to (x, z) to solve for β and α, and finally use Equation (1) to convert the result back into predicted values.

For the transformed data (x_i, z_i), we establish a simple linear regression model:

According to the least squares method, the parameter estimators are:

z ^ = β ^ x + α ^ which yields the regression equation ([3]).

To evaluate the model’s predictive performance for the original turbidity values y, we use the following three indicators:

Mean absolute error:

Root mean square error:

Coefficient of determination:

where y ^ i is the predicted value obtained by inverse transformation using Equation (1), and y ¯ is the observed mean value ([5]).

This research uses simulated data to carry out the modeling experiment. Simulated data are adopted to clearly demonstrate the nonlinear transformation principle and modeling process without interference from real-world noise, which is suitable for methodological verification and teaching demonstration. Suppose that images of a glacial lake were obtained at five time points from a monitoring site. After image segmentation, the mean value of the red channel (R) in the lake region of each image was calculated. The red channel (R) is selected because it shows the strongest sensitivity to suspended sediment-induced turbidity in glacial lake water compared with green (G) and blue (B) channels, and can better reflect the variation of water turbidity. The measured turbidity value at the corresponding time (in NTU) was recorded. The data are shown in Table 1. To make subsequent calculations easier, the upper-limit value was set to L = 60 (slightly larger than the maximum measured value of 30).

Logit transformation: for each y i , compute z i = l n ( y i / ( L − y i ) ) , where L = 60.

Linear regression: take R as the independent variable x and z as the dependent variable, and apply summation using Equations (4) and (5) to compute β ^ and α ^ .

Inverse prediction transformation: for any R, first compute z ^ , then substitute it into Equation (1) to obtain the turbidity prediction y ^ = L / ( 1 + e { − z ^ } ) .

Model evaluation: compute MAE, RMSE, and R² using Equations (6)-(8).

Table 1.Original data for the simulated experiment.

The original data were first subjected to a Logit transformation (with L = 60), and the results are shown in Table 2.

Table 2.Results of the logit transformation.

The required statistics are calculated as follows:

Substituting into Equations (4) and (5):

β ^ = 0.02708

α ^ = −4.3570

Therefore, the linear regression equation is:

The model parameters are statistically significant (p < 0.05), indicating that the red channel mean value has a significant linear effect on the Logit-transformed turbidity.

Substituting each R into Equation (9) gives z ^ , and then the predicted turbidity is calculated through y ^ = 60 / ( 1 + e { − z ^ } ) . The results are shown in Table 3.

Table 3.Prediction results and residual analysis.

MAE = 0.292 (NTU);

RMSE = 0.345 (NTU);

R² = 0.996.

Model effectiveness: R² = 0.996 indicates that the model explains 99.6% of the variation in turbidity, and MAE = 0.292 NTU shows that the average prediction error is only 0.292 units, indicating relatively high predictive accuracy.

Advantages of logistic regression: Compared with direct linear regression (directly modeling y against R without transformation, R² = 0.992, MAE = 0.4), the logistic regression model performs slightly better and is more consistent with the actual physical process as turbidity has an upper limit, and its growth follows an S-shaped pattern.

Meaning of the parameter: β = 0.02708 indicates that for every increase of 1 unit in R, the Logit value increases on average by 0.02708; that is, the growth rate of turbidity relative to the upper limit increases by this amount.

Limitations: In this experiment, L = 60 was chosen on the basis of an empirically determined regional upper limit. If L is set too high, the model will underestimate turbidity at high values; if L is set too low, the model will overestimate turbidity at high values, leading to systematic bias at the extremes of the turbidity range. In practical applications, optimization methods can be used to determine the optimal L.