Method for retrieving inorganic suspended matter and vegetation depth in optically shallow waters of submerged vegetation substrates

By constructing an optical shallow water simulation dataset and a multi-stage random forest model, the problem of inaccurate inversion of submerged vegetation in optical shallow water areas was solved, and high-precision inversion of vegetation depth and inorganic suspended matter concentration was achieved, thus improving the scientificity and effectiveness of lake ecological environment monitoring.

CN122224321BActive Publication Date: 2026-07-14NANJING INST OF GEOGRAPHY & LIMNOLOGY

Patent Information

Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
NANJING INST OF GEOGRAPHY & LIMNOLOGY
Filing Date
2026-05-21
Publication Date
2026-07-14

AI Technical Summary

Technical Problem

Existing technologies neglect the influence of the substrate in the shallow water zone of submerged vegetation, resulting in inaccurate vegetation depth and inorganic suspended matter inversion, insufficient physical constraints, and inadequate generalization ability and interpretability.

Method used

An optical shallow water simulation dataset based on a radiative transfer model was constructed. By using a random forest model combined with the two-flow radiative transfer equation, vegetation depth and inorganic suspended matter concentration were inverted. The model was trained using multi-band remote sensing reflectance and diffuse attenuation coefficient to improve inversion accuracy and interpretability.

Benefits of technology

It improves the accuracy and robustness of inversion of vegetation depth and inorganic suspended matter in shallow water areas of submerged vegetation substrate, and provides scientific support for lake ecological environment monitoring.

✦ Generated by Eureka AI based on patent content.

Smart Images

  • Figure CN122224321B_ABST
    Figure CN122224321B_ABST
Patent Text Reader

Abstract

The present application relates to a method for retrieving inorganic suspended matter and vegetation depth in optically shallow waters with submerged vegetation substrate, based on a radiative transfer model to simulate the optical shallow water dataset with submerged vegetation as the substrate, based on the optical shallow water dataset to build and train a two-stage random forest model that integrates physical constraints, and with the help of the two-stream radiative transfer equation to realize the parameter connection between stages, forming an inversion model suitable for inorganic suspended matter concentration and submerged vegetation depth in the optical shallow water area. The method improves the precision of the biological optical parameter inversion model in the optical shallow water area, and provides more scientific and effective technical support for lake ecological environment monitoring and management.
Need to check novelty before this filing date? Find Prior Art

Description

Technical Field

[0001] This invention belongs to the field of remote sensing technology for lake water environment and water color parameter inversion, specifically involving a method for inverting inorganic suspended matter and vegetation depth in shallow water areas of submerged vegetation substrate. Background Technology

[0002] Optical shallow water refers to water bodies where the bottom has a detectable influence on the water surface radiance or remote sensing reflectance. Its formation is related not only to water depth but also to water transparency and the attenuation characteristics of various optically active components in the water. In lakes, submerged vegetation is widely distributed in shallow water areas. Due to its low reflectance and strong absorption characteristics, it significantly alters the spectral shape and amplitude of the water surface remote sensing reflectance, thereby increasing the uncertainty in water color parameter inversion.

[0003] Existing remote sensing inversion methods for optically shallow water mainly focus on sandy or coral reef bottom scenarios in nearshore waters, while research on optically shallow water areas with submerged vegetation in inland lakes remains limited. For lake water parameter inversion, many existing algorithms assume the measured area is optically deep water and ignore the contribution of the bottom sediment, leading to significant systematic biases in submerged vegetation areas.

[0004] On the other hand, although machine learning methods can establish nonlinear mapping relationships between water parameters and remotely sensed reflectance using a large number of samples, models based solely on empirical features generally lack clear physical interpretation capabilities, and their generalization ability and interpretability remain insufficient in optically shallow water areas where sediment has a significant impact. Therefore, there is an urgent need to propose a joint inversion method for vegetation depth and inorganic suspended matter in optically shallow water areas with submerged vegetation that can take into account both physical mechanism constraints and data-driven advantages. Summary of the Invention

[0005] The purpose of this invention is to overcome the problems of existing technologies in the optical shallow water zone of submerged vegetation substrate, such as ignoring the influence of substrate, inaccurate identification, insufficient physical constraints, and limited robustness of parameter inversion, and to provide a method for inverting inorganic suspended matter and vegetation depth in the optical shallow water zone of submerged vegetation substrate.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A method for inverting inorganic suspended matter and vegetation depth in shallow optical zones of submerged vegetation substrate, the method comprising:

[0008] Using a radiative transfer model, we simulated different combinations of inherent optical parameters and corresponding remote sensing reflectance in optically shallow water areas under submerged vegetation substrate conditions, and constructed an optically shallow water simulation dataset.

[0009] Using the multi-band remote sensing reflectance from the optical shallow water simulation dataset as input, the initial vegetation depth H1 and the diffuse attenuation coefficient K of the optical shallow water discrimination band are used.d As output, train the first random forest model;

[0010] The diffuse decay coefficient K output by the first random forest model d Using the initial vegetation depth as the independent variable, the water column reflectivity R in the discrimination band is solved based on the two-stream radiative transfer equation. w And build R w / H1 is used as a feature parameter;

[0011] With the R w / H1, Diffuse attenuation coefficient K d The second random forest model is trained by taking multi-band remote sensing reflectance as input and the corresponding inorganic suspended matter concentration and vegetation depth as output.

[0012] Remote sensing data covering the study area was acquired, and the optical shallow water area of ​​the study area was extracted based on the remote sensing data. Based on the remote sensing reflectance parameters of the optical shallow water area, the corresponding multi-band remote sensing reflectance was used as input, and the inorganic suspended matter and vegetation depth of the optical shallow water area were inverted using the first random forest model and the second random forest model.

[0013] In some embodiments of the present invention, the radiative transfer model adopts the Hydrolight model; the simulation uses optical parameter models for pure water, algal particles, colored soluble organic matter, and non-algal particles.

[0014] In some embodiments of the present invention, the simulation parameters of the optical shallow water simulation dataset include at least chlorophyll a concentration and the absorption coefficient α of colored soluble organic matter at 440 nm. g (440), inorganic suspended matter concentration SPIM and vegetation depth;

[0015] The chlorophyll a concentration, a g (440) The parameter ranges for SPIM and vegetation depth are set based on the measured results of the study area.

[0016] In some embodiments of the present invention, when setting parameters, a step size is preset based on the parameter range, and the step size is encrypted for the typical parameter range of the study area.

[0017] In some embodiments of the present invention, the optical shallow water discrimination band is determined based on the following method:

[0018] Based on the simulation results of the radiative transfer model, the depth at which the remote sensing reflectance of each band reaches a stable value with the vegetation depth is obtained, and the band with the stable depth in the middle position is selected as the optical shallow water discrimination band.

[0019] In some embodiments of the present invention, the multi-band remote sensing reflectance is a combination of different single-band remote sensing reflectances;

[0020] The wavelength of the band selected for the single-band remote sensing reflectance is greater than the wavelength of the discrimination band.

[0021] In some embodiments of the present invention, the multi-band remote sensing reflectance does not include the blue light band.

[0022] In some embodiments of the present invention, the optical shallow water zone is based on Rayleigh-corrected reflectivity R. rc extract;

[0023] The multi-band remote sensing reflectance is the water surface remote sensing reflectance R. rs .

[0024] In some embodiments of the present invention, the extraction process of the optical shallow water zone is as follows:

[0025] The floating leaf / emergent vegetation identification index (FAVI) was calculated based on Rayleigh-corrected reflectance, and submerged vegetation and open water areas were extracted by combining the FAVI index and threshold segmentation method.

[0026] Submerged vegetation identification indices SVSI1 and SVSI2 are calculated for submerged vegetation and open water areas, and the submerged vegetation dominant region is extracted by threshold segmentation method; the submerged vegetation dominant region is the region composed of pixels that simultaneously satisfy the condition that SVSI1 index is less than a first threshold and SVSI2 is less than a second threshold.

[0027] In some embodiments of the present invention, the method further includes post-processing based on the optical shallow water area extraction results, including:

[0028] Reclassify isolated regions with fewer than a first preset threshold number of connected pixels; and,

[0029] For the extracted optical shallow water area, pixels with fewer than a second preset threshold number of effective pixels in an n×n neighborhood are removed; n is the preset neighborhood side length.

[0030] This invention constructs a simulated dataset of submerged vegetation substrate optical shallow water to determine the discrimination band applicable to this scenario; then, it introduces the physical parameters obtained by solving the radiative transfer equation into a two-stage machine learning model, thereby improving the accuracy and interpretability of the bio-optical parameter inversion model for optical shallow water areas, and providing more scientific and effective technical support for lake ecological environment monitoring and management. Attached Figure Description

[0031] The accompanying drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component shown in the various figures may be denoted by the same reference numeral. For clarity, not every component is labeled in each figure. Embodiments of various aspects of the invention will now be described by way of example and with reference to the accompanying drawings, wherein:

[0032] Figure 1 This is a flowchart of the two-stage random forest model construction process.

[0033] Figure 2 This is a display map showing the optical shallow water extraction results and the distribution of validation sample points in the study area on August 1, 2020; where (a) is the optical shallow water extraction results in the study area; and (b) is the distribution of validation sample points.

[0034] Figure 3 This is a schematic diagram of the optical shallow water extraction results in the study area during different seasons in 2023; (a) is the extraction result on January 18; (b) is the extraction result on April 8; (c) is the extraction result on August 21; and (d) is the extraction result on November 19.

[0035] Figure 4 This is a comparison chart of the model-calculated and measured values ​​of vegetation depth H in the optical shallow water area; where (a) is the calculation result on December 8, 2024; (b) is the calculation result on August 25, 2025; (c) is a comparison of the calculation result and the measured result at the sampling point on December 8, 2024; and (d) is a comparison of the calculation result and the measured result at the sampling point on August 25, 2025.

[0036] Figure 5 This is a comparison chart of the model calculation values ​​and measured values ​​of SPIM in the shallow water area; where (e) is the calculation result on December 8, 2024; (f) is the calculation result on August 25, 2025; (g) is a comparison of the calculation results and measured results of the sampling points on December 8, 2024; and (h) is a comparison of the calculation results and measured results of the sampling points on August 25, 2025.

[0037] Figure 6 The images show the results of the optical shallow water area extraction and its SPIM and H inversion. (a), (d), and (g) are the shallow water area extraction, SPIM inversion, and H inversion results on August 11, 2022, respectively; (b), (e), and (h) are the shallow water area extraction, SPIM inversion, and H inversion results on August 21, 2023, respectively; and (c), (f), and (i) are the shallow water area extraction, SPIM inversion, and H inversion results on August 30, 2024, respectively.

[0038] In the aforementioned Figures 1-6, the coordinates, symbols, or other representations expressed in English are all well-known in the field and will not be described again in this example. Detailed Implementation

[0039] To better understand the technical content of the present invention, specific embodiments are described below in conjunction with the accompanying drawings.

[0040] Various aspects of the invention are described in this disclosure with reference to the accompanying drawings, in which numerous illustrative embodiments are shown. The embodiments of this disclosure are not necessarily defined to include all aspects of the invention. It should be understood that the various concepts and embodiments described above, as well as those described in more detail below, can be implemented in any of many ways, because the concepts and embodiments disclosed herein are not limited to any particular implementation. Furthermore, some aspects of the invention disclosed may be used alone or in any suitable combination with other aspects of the invention disclosed.

[0041] Example 1

[0042] This embodiment uses a freshwater lake A as an example to further illustrate the technical solution of the present invention. This freshwater lake contains a large area of ​​optically shallow water, and the optical characteristics of these areas are significantly affected by the bottom vegetation, exhibiting optical response characteristics different from other areas of the lake. Therefore, this area has significant research value in areas such as water body optical parameter inversion and aquatic vegetation remote sensing identification.

[0043] The inorganic suspended matter and vegetation depth in the optical shallow water zone of the submerged vegetation substrate of this freshwater lake were retrieved using MSI satellite data, as follows:

[0044] (1) Data acquisition;

[0045] Multiple sampling points were selected in the lake, and water spectra were collected using a spectrometer to obtain the measured R values ​​at the lake sampling points. rs Data was collected using a spectrometer with a wavelength range of 350-1050 nm and a spectral resolution of 1 nm. At sampling points, a telescopic measuring rod was used to simultaneously measure vegetation depth, with the bottom of the rod contacting the surface of submerged vegetation and measuring its distance from the water surface. Submerged vegetation was collected from the bottom of the lake during sampling, and its reflectance data R(λ) was measured using the spectrometer. Water samples were collected from the surface layer of the lake (approximately 30 cm), preserved at low temperature and in the dark, and filtered on the same day. In the laboratory, the total suspended solids (SPM) concentration and the inorganic suspended solids concentration were measured using a gravimetric method.

[0046] Sampling and measurements should ideally be taken during the day, in sunny or partly cloudy weather with an average wind speed of less than 3 m / s. Since the growth status of submerged vegetation varies across seasons, sampling can be conducted in different seasons to evaluate measurement results under different optical conditions.

[0047] Download Sentinel-2 MSI Level-1C data products (Top Atmospheric Reflectance (TOA)) covering Freshwater Lake A from 2016 to 2024 from the Copernicus Data Center. Select 107 images that fully cover Freshwater Lake A under clear weather conditions. Atmospheric correction was performed on the downloaded data products using the Dark Spectral Fitting (DSF) method based on dark pixels on the Acolite platform, outputting Rayleigh-corrected reflectance R. rc Image and remote sensing reflectance R rs image.

[0048] (2) Optical shallow water to R rs Analysis of Influence Patterns

[0049] Taking the submerged vegetation distribution area in the eastern part of freshwater lake A as an example, a shallow water optical simulation dataset of submerged vegetation substrate was constructed using Hydrolight-Ecolight 5.0. The simulation employed a four-component intrinsic optical parameter model: pure water, algal particles, colored soluble organic matter, and non-algal particles; the substrate reflectance was based on the average reflectance of submerged plants built into the model. The simulation wavelength range was 400–900 nm.

[0050] Determine the concentration of chlorophyll a (Chl-a) and the absorption coefficient of colored soluble organic matter at 440 nm (a g (440)) and the typical range of variation of inorganic suspended matter (SPIM) concentration, and on this basis, set the range and interval of simulation parameters.

[0051] In this embodiment, a standard parameter range is set, and sampling is further intensified within the typical water body parameter range of freshwater lake A to enhance the model's ability to represent common optical conditions of the target lake. The vegetation depth range is segmented according to the parameter value strategy: within the standard range, the vegetation depth is set to 0.1–2.0 m; within the intensified range for typical conditions of freshwater lake A, the vegetation depth range is further expanded to 1.0–2.5 m. In both cases, a step size of 0.1 m is used, thus obtaining simulation results for multiple different vegetation depth scenarios under the same optical parameter conditions. Smaller vegetation depth intervals help capture subtle response characteristics of remote sensing reflectance signals as vegetation depth changes. The main simulation parameter settings are shown in Table 1.

[0052] Table 1. Hydrolight Main Simulation Parameter Settings

[0053]

[0054] In this embodiment, 32,141 optical shallow water simulation samples of submerged vegetation substrate were ultimately formed for subsequent model training.

[0055] Furthermore, this embodiment also sets up deep-water data under the same conditions (the difference from the submerged vegetation substrate optical shallow-water data is that it does not include submerged vegetation substrate). Analysis of the simulation results shows that submerged vegetation substrate will cause the shallow-water area R... rs It decreases in the visible light band and increases in the near-infrared band; as vegetation depth increases, R in shallow water areas... rs Gradually moving towards optical deep-sea R rs The convergence indicates that the influence of the substrate weakens as the propagation path lengthens. Further sensitivity comparisons of different optically active components revealed that SPIM has a lower effect on R... rs The impact of stability depth is most significant, therefore SPIM is one of the main inversion targets.

[0056] (3) Model building

[0057] like Figure 1 As shown, by analyzing different wavelengths R in the range of 400–900 nm rs The variation in stable depth was determined based on the stable depth of each band. The 560nm band (the green light band centered at 560nm), where the stable depth is in the middle, was selected as the optical shallow water discrimination band to characterize the influence of submerged vegetation substrate. The R value in the simulation data was used as the basis for this determination. rs (560), R rs (665), R rs (705), Rrs(740), R rs (783), R rs (842) and R rs Different permutations and combinations of (865) are used as input features of the model, with initial depths H1 and K. d (560) Using these as output features, a first-stage random forest model (RF1) is trained. The output accuracy of different combinations is compared, and the optimal combination of input features is selected as the input parameters for the final first-stage model. In this embodiment, after comparing different combinations of input bands, it was found that when all seven bands were used as input features, the constructed model exhibited the highest overall prediction accuracy.

[0058] After obtaining the initial depths H1 and K d (560) After that, the water column reflectivity R at 560 nm is solved based on the two-stream radiative transfer theory. w (560), as follows:

[0059] R w (560) = Q×R rs (560) / 0.54 - R b (560)×exp[-2K] d (560)H1]

[0060] Where Q is the light field distribution parameter, which is taken as 4.0 in this embodiment, referring to existing literature; R b (560) represents the reflectance of the submerged vegetation substrate at 560 nm, which is taken as 0.07 in this embodiment. Subsequently, R0 was constructed. w (560) / H1 features are used to characterize the reflection contribution per unit water column.

[0061] The R-value will be solved by combining the first-stage random forest model and the second-flow radiative transfer theory. w (560) / H1 and K d (560), and the final R obtained in the first stage of model training. rs The input parameters are used as the input to the model, and the corresponding simulated data of SPIM and depth H are used as the output to train the second-stage random forest model (RF2).

[0062] In this embodiment, the hyperparameters of the random forest model are preferably set as follows: n_estimators=250, max_depth=None, max_features=sqrt, and the ratio of training set to test set is 7:3. To evaluate the robustness of the model under noisy conditions, the model performance can be evaluated again after applying a random perturbation of no more than 10% to the input features of the test set.

[0063] (4) Satellite applications

[0064] 1. Extracting the optical shallow water zone of lakes based on remote sensing data;

[0065] Using the R obtained in step (1) rc Data, calculating the Floating Leaf / Emerging Vegetation Identification Index (FAVI) based on the Sentinel-2 red-edge band:

[0066]

[0067] In the formula, It is the third red-edge band reflectivity of Sentinel-2, with a center wavelength of 783nm.

[0068] The FAVI index combined with the OTSU method was used to perform threshold segmentation on the image to distinguish emergent and floating-leaved vegetation from submerged vegetation and water. The extracted emergent and floating-leaved vegetation was masked, leaving only the pixels of submerged vegetation and water.

[0069] Distinguish between submerged vegetation areas and other water areas based on the pixels of the preserved submerged vegetation and water:

[0070] Calculate the spectral sensitivity indices SVSI1 and SVSI2 for submerged vegetation for the remaining pixels:

[0071]

[0072]

[0073] In the formula This represents the brightness index after the tassel cap is transformed. This represents the greenness index after the tassel cap is changed; This is the green light reflectance of Sentinel-2, with a center wavelength of 560nm. Considering that the blue light band in the visible light spectrum is greatly affected by atmospheric correction, and that water bodies have a stronger absorption capacity for the red light band than the green light band, the green light band is more effective in identifying submerged vegetation in freshwater lake area A.

[0074] Obtain using the OTSU method threshold ,as well as threshold , will simultaneously satisfy and The pixels were identified as submerged vegetation optical shallow water pixels; isolated regions with fewer than 10 connected pixels were reclassified; and pixels with fewer than 5 effective pixels in a 3×3 neighborhood were further removed to reduce the impact of edge noise and misclassification.

[0075] The results of optical shallow water extraction in the study area in 2020 and the distribution of validation sample points are as follows: Figure 2 As shown; some optical shallow water extraction results in 2023 are as follows. Figure 3 As shown.

[0076] 2. Inversion of optical shallow water area SPIM and vegetation depth H

[0077] The extracted submerged vegetation optical shallow water area R rs Images, and R w (560) / H1 and K d (560) Input the second-stage random forest model to obtain the inversion results of SPIM and vegetation depth H. Figure 4 , Figure 5 , Figure 6 R here w (560) / H1 and K d (560) is obtained by referring to step (3) through the optical shallow water area R of submerged vegetation. rs The first-stage random forest model trained from the image input is used to solve the problem by combining it with the theory of second-order radiative transfer.

[0078] In this embodiment, the accuracy of the inversion results is verified using the measured data collected in step (1), and the model accuracy under different combinations of input parameters is compared. The results are shown in Table 2.

[0079] Table 2 Model accuracy for different combinations of input parameters

[0080]

[0081] It can be seen that, by simultaneously introducing the supplementary parameter K... d (560), R w The second-stage model (560) / H1 significantly improves the inversion accuracy. On a test set with 10% random error, the prediction accuracy R² of the second-stage model for SPIM and vegetation depth H can reach approximately 0.9900 and 0.8323, respectively, demonstrating high accuracy and robustness.

Claims

1. A method for inverting inorganic suspended matter and vegetation depth in shallow optical zones of submerged vegetation substrate, characterized in that, The method includes: Using a radiative transfer model, we simulated different combinations of inherent optical parameters and corresponding remote sensing reflectance in optically shallow water areas under submerged vegetation substrate conditions, and constructed an optically shallow water simulation dataset. Using the multi-band remote sensing reflectance from the optical shallow water simulation dataset as input, the initial vegetation depth H1 and the diffuse attenuation coefficient K of the optical shallow water discrimination band are used. d As output, train the first random forest model; The diffuse decay coefficient K output by the first random forest model d Using the initial vegetation depth as the independent variable, the water column reflectivity R in the discrimination band is solved based on the two-stream radiative transfer equation. w And build R w / H1 is used as a feature parameter; With the R w / H1, Diffuse attenuation coefficient K d The second random forest model is trained by taking multi-band remote sensing reflectance as input and the corresponding inorganic suspended matter concentration and vegetation depth as output. Remote sensing data covering the study area was acquired, and the optical shallow water area of ​​the study area was extracted based on the remote sensing data. Based on the remote sensing reflectance parameters of the optical shallow water area, the corresponding multi-band remote sensing reflectance was used as input, and the inorganic suspended matter and vegetation depth of the optical shallow water area were inverted using the first random forest model and the second random forest model.

2. The method according to claim 1, characterized in that, The radiative transfer model adopted is the Hydrolight model; the simulation uses optical parameter models for pure water, algal particles, colored soluble organic matter, and non-algal particles.

3. The method according to claim 1 or 2, characterized in that, The simulation parameters of the optical shallow water simulation dataset include at least chlorophyll a concentration and the absorption coefficient of colored soluble organic matter at 440 nm. g (440), inorganic suspended matter concentration SPIM and vegetation depth; The chlorophyll a concentration, a g (440) The parameter ranges for SPIM and vegetation depth are set based on the measured results of the study area.

4. The method according to claim 3, characterized in that, When setting parameters, preset the step size based on the parameter range, and refine the step size for the typical parameter range of the study area.

5. The method according to claim 1, characterized in that, The optical shallow water discrimination band is determined based on the following method: Based on the simulation results of the radiative transfer model, the depth at which the remote sensing reflectance of each band reaches a stable value with the vegetation depth is obtained, and the band with the stable depth in the middle position is selected as the optical shallow water discrimination band.

6. The method according to claim 1, characterized in that, The multi-band remote sensing reflectance is a combination of different single-band remote sensing reflectances; The wavelength of the band selected for the single-band remote sensing reflectance is greater than the wavelength of the discrimination band.

7. The method according to claim 1, characterized in that, The multi-band remote sensing reflectance does not include the blue light band.

8. The method according to claim 1, characterized in that, The optical shallow water zone is based on Rayleigh-corrected reflectivity R. rc extract; The multi-band remote sensing reflectance is the water surface remote sensing reflectance R. rs .

9. The method according to claim 1, characterized in that, The extraction process for the optical shallow water zone is as follows: The floating leaf / emergent vegetation identification index (FAVI) was calculated based on Rayleigh-corrected reflectance, and submerged vegetation and open water areas were extracted by combining the FAVI index and threshold segmentation method. Submerged vegetation identification indices SVSI1 and SVSI2 are calculated for submerged vegetation and open water areas, and the submerged vegetation dominant region is extracted by threshold segmentation method; the submerged vegetation dominant region is the region composed of pixels that simultaneously satisfy the condition that SVSI1 index is less than a first threshold and SVSI2 is less than a second threshold.

10. The method according to claim 9, characterized in that, The method further includes post-processing based on the optical shallow water area extraction results, including: Reclassify isolated regions with fewer than a first preset threshold number of connected pixels; and, For the extracted shallow water area, remove n × n The number of valid pixels in the neighborhood is less than the second preset threshold. n The default neighborhood edge length.