Application of Inverse Mapping for Automated Determination of Normalized Indices Useful for Land Surface Classification

Precise surface classification is essential for glacial health monitoring, where normalized indices have traditionally been used. These indices are created empirically for a specific sensor. The transferability of these indices to other sensors can be affected by differences in spectral and spatial resolution. Thus, it is essential to evaluate the transferability of an index before applying it to a new sensor to ensure accuracy and reliability. However, as the number of satellites, sensors, and observation bands increases, there is a need for automated methods for determining application-specific normalized indices. In this article, we propose using all the bands of multispectral optical sensors to generate multiple normalized indices and determining application-specific indices using inverse mapping. We use these normalized indices for pixel-by-pixel surface classification using neural networks. First, we employ all the bands for generating normalized indices and then eliminate low-spatial-resolution bands to evaluate classification performance by using only high-spatial-resolution indices. We apply this method to a glacial region and observe 81.98% and 84.81% overall accuracy compared to the ground truth data for the two classifications, respectively. We then apply inverse mapping dynamics to the classification results to determine prominent indices useful for glacier classification. The results show that although some of the determined indices are not traditional indices, they are still useful for classification due to the relative differences between various land types. The proposed method has the potential to automate normalized index determination, thereby eliminating the need for empirical band assessment methods and making the index development process more efficient.


I. INTRODUCTION
M OUNTAIN glaciers are highly sensitive indicators of climate change [1], [2]. Thus, a detailed and accurate classification of the size of a glacier is an important step in order to understand their overall health and other applications such as water resource management and early prevention of glacial lake outburst floods (GLOFs) [3], [4], [5]. The rugged terrain and remote locations of glacierized regions limit the applicability of traditional in situ observation methods [6], [7]. However, the recent increase in the availability of multispectral optical remote sensing data with high resolution spatially and temporally allows frequent coverage of glacial regions, even in rugged terrain [6].
In most studies focusing on optical sensor data, index-based methods are used, wherein normalized indices are calculated from various spectral bands of optical sensor data and used to analyze changes. By determining the normalized difference of the spectral bands, it is possible to highlight specific characteristics and analyze trends and patterns that may not be apparent when looking at the raw data alone. Traditionally, normalized indices are obtained by observing the reflectance of the bands, determining target-specific high and low reflectance bands, and calculating their normalized differences.
Over the past three decades, several normalized indices have been suggested for different applications, and some of these indices are widely used for glacial applications. Among the commonly used indices is the Normalized Difference Snow Index (NDSI) [8], which was initially developed for Moderate Resolution Imaging Spectroradiometer to distinguish between snow and ice by utilizing the green and the shortwave infrared band. The Normalized Difference Vegetation Index [9] is also utilized often as it can indicate the presence of vegetation on the glacier surface. In addition, for glacial lake monitoring, the Normalized Difference Water Index (NDWI) [10] is often used and is useful to identify the presence of water in glacial lakes and to monitor changes in lake extent over time. In addition to NDWI, the Modified Normalized Difference Water Index (MNDWI) [11], which uses the same spectral bands as NDSI, has been widely used in glacial lake monitoring.
Glacierized terrains are spectrally complex regions and comprise different land covers with similar spectral characteristics [12]. The spectral similarity between glaciers and the surrounding material makes many of the conventional methods ineffective for identifying glaciers [13]. To address this issue, various new normalized difference indices have been proposed by combining different bands to enhance the distinctions between glaciers and the spectrally similar surrounding terrain [14]. A recent work [15] proposed methods for distinguishing between snow cover glaciers (SCGs) and water bodies. They introduced two indices, namely, the NDWI with no SCG information for extracting lake water and the NDSI with no water information to highlight SCG and to suppress the influence of lake water. The newly developed indices were proven successful in distinguishing between two land types that exhibited similar spectral features. However, their reliance on specific thresholds poses limitations to their application beyond the study area. In order to apply these indices in new areas, it is necessary to determine new thresholds, which in turn requires the user's judgment.
Moreover, in the context of water body extraction, it has been observed that indices such as NDWI and MNDWI were initially proposed and tested on a specific sensor, and the accuracy of their results may differ when applied to different sensors. Given that the Sentinel-2 sensor shares similar spectral and spatial characteristics with the Landsat series, it is expected that water indices developed on Landsat could presumably be applied to Sentinel-2 data [16]. However, despite the efficacy of these indices in land-type extraction of their respective sensors, a slight change in the spectral range of the corresponding band of another sensor could alter the results [17]. Moreover, the transferability of indices could also be influenced by the difference in spatial resolution between two sensors [18]. It, thus, becomes imperative to carefully evaluate the transferability of an index to a new sensor before applying it to ensure the accuracy and reliability of the results. Thus, as the number of sensors and observation bands increases, it is essential to develop automated methods for the determination of application-specific indices.
In the last few decades, researchers have developed various automated and semiautomated methods to classify different types of remote sensing satellite imagery [19], [20], [21], and recently, a lot of researchers have also focused on making explainable networks to understand the effect of the inputs on the classification results [22], [23]. In the realm of neural network explainability for remote sensing applications, inverse mapping has proven effective in giving reliable and convincing results [24].
In this article, we propose the use of all the combinations of raw band data of Sentinel-2 optical sensor for generating normalized indices. A fully connected feedforward neural network is employed for the pixel-by-pixel classification of a glacial region, and the efficacy of the indices for the classification task is evaluated. In addition to the classification, inverse mapping is used to determine the most significant indices useful for characterizing a particular land type.
This article first focuses on the proposed method in Section II, where we elaborate on the normalized index generation for land surface classification. In addition, prominent normalized index identification by using inverse mapping is introduced. Section III presents the experiments and results of the application of the proposed method on the dataset for the Imja glacier region. Section IV provides a detailed discussion on the results. Finally, Section V concludes this article.

A. Normalized Index Generation
In general, a normalized index is calculated as where B i and B j denote the spectral bands i and j, respectively; ρ band i denotes the reflectance of the corresponding band i, while ρ band j denotes that of band j.
In this experiment, a normalized index generator is modeled as follows. The input to the normalized index generator is N spectral bands of a particular sensor. It generates I = N C 2 combinations of N spectral bands in the form of (1). Consequently, the I combinations are channeled as inputs to the neural network. For land surface classification, each input terminal of the network is fed pixel values of a specific normalized index, as detailed in the following subsection.

B. Land Surface Classification
In this experiment, a feedforward neural network is employed for generating the pixel-by-pixel classification map. In this study, our aim is to achieve high-resolution classification focusing on the reflectance values of individual pixels. Unlike convolutional neural networks that diminish data resolution through pooling, the fully connected neural network introduced in this experiment directly handles the physical values instead of identifying texture or shape characteristics from the images and ensures that the spatial resolution remains uncompromised.
As shown in Fig. 1, the I normalized indices generated by the normalized index generator are used as inputs. This feedforward neural network is a fully connected single-hidden layer neural network consisting of an input terminal layer, a single hidden layer, and an output neuron layer.
Considering x = [x 1 x 2 . . . x I ] T ([·] T : transpose) and y = [y 1 . . . y J ] T to be the input and hidden signals, respectively, to be the biases for the hidden and output layers, respectively, and W 1 and W 2 to be the input-hidden weights and the hidden-output weights obtained after a learning phase, respectively, the values of z, where z = [z 1 z 2 . . . z K ] T , at the output layer of the neural network can be calculated as This network employs the modified logarithmic activation function proposed in [25], working componentwise and is defined as . (3)

C. Normalized Index Identification
Inverse mapping is a dynamics that determines the most prominent contributing input of a forward processing neural network by accessing the signal flow in the network. To gain insight into the inverse mapping process, it is beneficial to examine how a neural network identifies a winning neuron. This can be achieved by focusing on the role of the weights in the forward processing network. As depicted as Fig. 1, the input signals progress in a forward direction through W 1 and W 2 matrices. The respective weight values determine the "conductivity" of the neural connections for these signals, resulting in a set of outputs. Then, we can consider a reciprocal signal flow by taking transposed matrices W T 1 and W T 2 as the inverse signal propagation. No additional learning or weight optimization is necessary for this inverse mapping process [25]. Fig. 2 shows the inverse mapping process. In this process, the value obtained at the winning output node of the forward processing network is fed to the same node as the input of the inverse mapping, with all other nodes fed zeros as inputs. This approach effectively suppresses any potential influence that nonprominent classes might have. The network identifies the most significant features by tracking the backward signal flow from the winning node [25]. The following equation governs the inverse mapping: where z = [0 0 . . . zk . . . 0] T is the modified output values of the forward processing network fed as an input to the inverse mapping network, x is the output of the inverse mapping network, and the inverse activation function f −1 is defined as In this experiment, the inverse mapping identifies which of the input normalized indices are decisive and influential to the decision making of the neural network. The analysis is done pixel by pixel and class by class, wherein the output classes are correlated with the relevant input features. The inverse mapping results are displayed by using box-whisker plots. The results are displayed for all those pixels of a specific class that the neural network is confident of in the selected area of interest (AOI).
The proximity of the individual box plot to either −1 or +1 level represents high significance of the corresponding input feature [25]. The proximity of a feature to +1 (positively prominent feature) implies that the said feature positively influences the decision making of the feedforward neural network. On the other hand, the proximity of a feature to −1 (negatively prominent feature) implies that the said feature oppositely influences the decision-making process.

A. Study Area
Imja glacier is situated in the Khumbu region of eastern Nepal (27 • 54 17 N, 86 • 55 31 E). Imja glacier has two branch tributary glaciers: the Lhotse Shar in the northeast and the Ambulapcha glacier in the south [26]. The meltwater from Imja and Lhotse glaciers creates the Imja Tsho glacial lake, which is considered to be at high risk for GLOFs. Due to its close proximity to the Mt. Everest base-camp trekking route, the Imja glacier has been the focus of comprehensive research using both on-site and remote sensing techniques, making it one of the most extensively studied glaciers [27], [28], [29], [30]. Fig. 3 shows the AOI for this study. The AOI also includes parts of the Lhotse glacier and the Ambulapcha Tsho lake, a circular basin. Unlike the Imja Tsho, this lake has no visible watershed and the water discharges via springs [31]. Despite extensive research on Imja Tsho lake, there is a scarcity of literature regarding Ambulapcha Tsho lake. Given its close proximity to a glacier that is at risk, it is imperative to conduct in-depth study on Ambulapcha Tsho. Since the lake is not fed by a glacier, hereafter we consider it as a freshwater lake basin.
In addition, the topographic complexity, such as the presence of steep slopes in the AOI, introduces the effect of topographic shadows. Some of these shadow-contaminated regions are indicated in Fig. 3 by red outlines.

B. Dataset
The AOI dataset was obtained from the Sentinel-2 multispectral mission of the European Space Agency. Glacier inspections are usually carried out toward the end of the ablation season, as imagery obtained during this period can be useful for determining the end-of-summer snow-line altitude [6]. Accordingly, the acquisition of the dataset was done on October 14, 2021, which was toward the end of the ablation season.
The Sentinel-2 dataset was obtained as Level 1C images for all the bands, and the Sen2cor algorithm was applied to produce Level 2A images of all the bands, with each pixel having a spatial size of 10 m × 10 m. This ensures that each pixel of the corresponding bands represents the same geographical location when collocation is performed. Fig. 4 shows the ground truth map of the AOI. The AOI consists of 769 × 973 = 748 237 pixels. We classified each of the pixels in the AOI into five different land types: 1) snow; 2) glacial debris; 3) rock; 4) glacial lake; and 5) freshwater lake.

C. Ground Truth
The ground truth data were mapped pixel by pixel manually by careful visual inspection of Sentinel-2 and Google Earth Pro imagery. Despite the meticulous manual mapping efforts, the possibility of errors cannot be completely ruled out, especially in regions that are subject to shadows or located on steep slopes.

D. Feedforward Network for Land Surface Classification
As mentioned in Section II-B, the normalized index generator is used to generate I combinations of N spectral bands. The normalized indices generated from the spectral bands are used 2) only high-spatial-resolution spectral bands (10 and 20 m). The architecture of the neural network consists of three layers: an input layer with I nodes, a single-hidden layer with 64 neurons, and an output layer with five neurons. Each neuron in the output layer represents one of the five distinct land cover classes. As shown in Fig. 1, the input layer of the neural network is fed with pixel-by-pixel values of the generated normalized indices. At the output layer, the network classifies each pixel as one of the five land cover types: snow, glacial debris, rock, glacial lake, and freshwater lake. Both the feedforward neural networks are trained and validated using a total of 2500 ground truth pixels (enclosed in red boxes in Fig. 4) of their respective input datasets. The training pixels were carefully chosen considering both ground truth data and visual inspection of the RGB composite image of the region. The pixel selection involved a detailed examination of the composite image to identify suitable pixels, while the ground truth data ensured the accuracy of the labeling. The training validation dataset is partitioned into training and validation sets in an 80:20 ratio.
1) Feedforward Network for Land Surface Classification Using All Spectral Bands: The raw Sentinel-2 data contain 13 bands, as shown in Table I. The normalized indices are generated by utilizing all 13 spectral bands, resulting in I = 13 C 2 = 66 bands in total.
The feedforward neural network's input layer consists of 66 nodes, and each node is fed with a normalized index's pixel  value. The hyperparameters of the network are empirically selected through a rigorous process of trial and error as follows. The learning rate is set at 10 −5 and the network is trained for 2500 epochs. Fig. 5(a) displays the learning curves of the network. At the end of the training process, the final training loss and validation loss are 0.0703 and 0.0737, respectively.
The feedforward neural network is employed for generating the classification map by using the 45 normalized indices generated by the normalized index generator. The hyperparameters of this network are also chosen empirically, the learning rate is set at 10 −5 , and the network is trained for 2500 epochs. Fig. 5(b) depicts the achieved training loss and validation loss, which are 0.2057 and 0.2017, respectively. Fig. 6(a) and (b) shows the classification maps generated by the neural networks with all resolution normalized indices and high-spatial-resolution normalized indices, respectively. A visual comparison of the two classification maps shows that both the maps have a good classification. However, it is worth noting that Fig. 6(a) exhibits a relatively coarse classification when compared to Fig. 6(b), which can be attributed to the inclusion of indices containing 60-m resolution bands.

E. Classification Results
The classification results indicate that both the regions are successful in accurately classifying snow-covered areas and distinguishing between different types of water bodies, including supraglacial ponds on glacial debris. In addition, both the regions are effective in differentiating between spectrally similar glacial debris and rock land types.
In addition, the results demonstrate the effectiveness of both the networks in classifying areas under shadow, which has been a long-standing challenge in the field of optical remote sensing. However, Fig. 6(b) exhibits that some misclassification still remains in regions located on mountain slopes.

F. Comparison With Ground Truth Data
The classification maps presented in Fig. 6(a) and (b) are compared pixel by pixel to the ground truth data depicted in Fig. 4. The classifications have also been quantitatively evaluated by using recall scores (RSs). Each individual RS has been calculated as

RS =
True positive True positive + False negative × 100. Fig. 7(a) shows the confusion matrix for the classification using all the normalized indices. The classification results exhibit an overall accuracy of 81.98%. Fig. 7(b) shows the confusion  matrix for the classification using high-spatial-resolution normalized indices only. The classification results exhibit an overall accuracy of 84.81%.
In both the confusion matrices, the most prominent misclassification arises from the incorrect identification of rock as glacial debris. This misidentification can be attributed to the similarity in spectral signatures or to limitations in the mapping of the ground truth data for these two land types, particularly in regions affected by shadows and steep slopes.

G. Normalized Index Identification Using Inverse Mapping
The classification results presented in Section III-E are utilized to identify the prominent input via inverse mapping. As mentioned in Section II-C, the inverse mapping results are displayed by using box-whisker plots, and the proximity of the individual box plot to either −1 or +1 level represents the significance of the corresponding input feature. If an index is close to +1 level, it indicates a positively prominent index, while if an index is close to −1, it is considered a negatively prominent index.
In addition, for ease of interpretation of inverse mapping results, normalized index maps of the input feature identified as prominent have also been provided. By analyzing the index maps, we identify two types of normalized indices: 1) absolute indices that independently denote a target region with a −1 or +1 value and 2) relative indices that rely on the relative differences between the land types.
1) Inverse Mapping for Classification Using All 66 Normalized Indices: Fig. 8 depicts the results of inverse mapping for  a) Snow and glacial lake: Fig. 8(a) and (d) shows that B 5 : B 9 is a prominent index for the detection of snow and glacial lake pixels and is calculated by (7). Based on the proximity to the index to −1 and +1 level, it can be noted that for the determination of the snow pixels, B 5 : B 9 is a negatively prominent index. However, for the determination of the glacial lake pixels, it is a positively prominent index.
The corresponding index map for B 5 : B 9 is shown in Fig. 9(a). Here, the glacial lake pixels are indicated by +1 value, while the snow pixels are determined by using their relative difference from the glacial lake pixels. Thus, B 5 : B 9 is considered to be a relative index. b) Debris and rock: Fig. 8(b) and (c) indicates that debris pixels and rock pixels also have a complementary relationship. The most significant index for both is the B 1 : B 3 index which is calculated as (8). B 1 : B 3 positively affects the neural network for the classification of debris pixels and negatively affects the neural network's classification of rock pixels.
The index map for B 1 : B 3 is shown in Fig. 9(b), where the rock regions are indicated by a −1 value, while the debris regions are indicated by a value closer to zero. Since this index is determined by the relative differences between two classes, it is classified as a relative index. c) Freshwater lake: From Fig. 8(e), the most prominent indices for freshwater lake pixels are determined to be B 6 : B 11 and B 6 : B 12 , given by (9) and (10), respectively. The corresponding index maps for B 6 : B 11 and B 6 : B 12 are shown in Fig. 9(c) and (d), respectively, where the freshwater lake pixels are indicated by a −1 value. The index maps clearly distinguish between the two water bodies. In these index maps, since the target area is indicated by a −1 value, both indices are considered to be negatively prominent absolute indices.

2) Inverse Mapping for Classification Using 45 High-Spatial-Resolution Normalized Indices:
The inverse mapping results using only high-spatial-resolution indices are given by Fig. 10, and Table III summarizes the indices that are found prominent.
a) Snow and glacial lake: Fig. 10 b) Debris and rock: From Fig. 10(b) and (c), B 2 : B 8A is found to be a positive index for glacial debris and a negative index for rock pixels. B 2 : B 8A is calculated in (15).
The corresponding map is shown in Fig. 12. The rock regions are denoted by a value of −1 in the maps, whereas the debris regions are represented by values closer to zero. Since the results of the inverse mapping of the debris class exhibit weakly positive values, only B 2 : B 8A is considered as a prominent normalized index and other normalized indices have been excluded from the analysis.
c) Freshwater lake: Similar to the results shown in Section III-G1c, the inverse mapping for the high-spatial-resolution freshwater lake classification also indicates the prominence of the B 6 : B 11 and B 6 : B 12 indices. In addition, B 3 : B 6 , B 3 : B 8 ,  (19).
The corresponding index maps for these indices are shown in Fig. 13. Based on the inverse mapping results and the index map, it is evident that B 6 : B 11 and B 6 : B 12 are negatively prominent indices, while B 3 : B 6 , B 3 : B 8 , B 4 : B 6 , and B 5 : B 6 are positively prominent indices. Since these indices clearly exhibit a value of +1 or −1 for the target region, they can be classified as absolute indices.

IV. DISCUSSION
This article focuses on determining normalized indices that are useful for identifying and classifying glacial facies. A normalized index generator and a fully connected feedforward neural network have been utilized to accomplish this goal. In addition, inverse mapping dynamics was applied to the classification results to determine prominent indices useful for glacier classification. It was found that the indices were successful in classifying the glacier region, and the inverse mapping gave reliable results for the determination of new indices. It was found that even though some of the determined indices are not traditional absolute normalized indices, they can still determine the classes by utilizing the relative differences between the various land types.
From the results of this study, it was found that, for an absolute index in the negatively prominent index's map, the target region pixels have a value <0. Additional experiments showed that a negatively prominent index B i : B j implies that its complementary index B j : B i is also a significant index, and the pixels in the target area in the index map have a value >0.
In addition, we were able to successfully identify numerous impactful indices for the classification of glacial facies. It is noteworthy that some of these indices were already being utilized before their significance was identified in this study. For instance, the B 3 : B 8 index/Freshwater Index-4 (FWI 4 ), which was significantly impactful for the high-resolution classification of freshwater lake, is calculated using the same bands as NDWI [10]. Similarly, the B 2 : B 4 index/Snow and glacial lake index-2 (SGI 2 ), determined to be a significant index for snow and glacial lake classification, employed the same formula as NDWI 1 [32], [33], [34].
Furthermore, our findings supported a recent study [17] that proposed the Sentinel-2 water index (SWI), by taking into account the influence of the vegetation-sensitive-red-edge bands of Sentinel-2. SWI was calculated by normalizing the VRE 1   III  PROMINENT INDICES DETERMINED FROM THE INVERSE MAPPING OF THE CLASSIFICATION OBTAINED BY USING ONLY 45 HIGH-SPATIAL-RESOLUTION  NORMALIZED INDICES band with the SWIR 2 band. In our study, we introduced the B 6 : B 11 index/Freshwater index-1 (FWI 1 ), which exhibited similarities to SWI as it involved normalizing the VRE 2 band with the SWIR 2 band. The identification of these relevant preexisting indices supports the generalization ability of the proposed method. Inverse mapping had previously shown efficacy in determining important features related to earthquake disaster assessment [35]. The effectiveness of inverse mapping in the current application is demonstrated by the fact that the indices identified by our method have already been adopted in practical applications. This highlights the potential of our method to be used as a valuable tool for identifying and developing new indices that could be of great value for classification tasks in various domains. By leveraging the relationships between different spectral bands, our method can effectively capture the  unique characteristics of different features, making it a powerful tool for extracting information from remotely sensed data.
Moreover, in order to further evaluate the impact of normalized indices on classification, additional experiments were conducted by using only raw Sentinel-2 band data as input. These experiments showed that the accurate classification of shadowed areas was not possible without the use of normalized indices. This is because the presence of shadows can cause significant variations in the reflectance values of different bands, leading to inaccurate classification results. By incorporating normalized indices that account for these variations, the effects of shadows on classification accuracy can be mitigated. This highlights the importance of using normalized indices in classification, especially in areas with shadows or other topographically challenging conditions that can affect the accuracy of classification results.
Our proposed method in this study has the potential to be extended to determine additional indices beyond those examined in this study. These could include indices based on methods such as simple band ratios or double differences, which have been shown to be effective in various applications [36], [37]. Therefore, the ability of our proposed method to identify effective indices, such as simple band ratios or double differences, could have broader applications in remote sensing and environmental monitoring.

V. CONCLUSION
In this article, we proposed the utilization of all the combinations of Sentinel-2 indices for land classification and the determination of useful indices. The effectiveness of this approach was demonstrated through its application in the classification of a glacial region, where indices prominent for glacial classification were identified by using inverse mapping. In addition, the method effectively classifies shadowed regions, which has been a long-standing challenge in optical remote sensing. Moreover, some of the indices determined to be effective by the proposed method are already being used in practical applications. Given the anticipated growth in the number of sensors and observation bands in the near future, this method has the potential to become an effective method for identifying application-specific indices.