# A New Digital Lake Bathymetry Model Using the Step-Wise Water Recession Method to Generate 3D Lake Bathymetric Maps Based on DEMs

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## Abstract

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## 1. Introduction

- Airborne observation: This method uses airborne gravity data to determine the lake bottom shape with the assumed density of sediment [11]. The advantage of this method is that it can generate the sediment layer due to the density variance; the disadvantage is the high cost of airborne equipment for large-scale observations [11].
- Passive imaging method: This method is widely applicable to shallow turbid water bodies by the different reflectances of different wave bands [12]. There are two main ways to establish the relationship between the depth and reflectance (i.e., the equation fitting method) [13] and the machine learning method, such as ANN (artificial neural network) [14]. However, a necessary condition for this method is that the water must be shallow enough; it does not work for large and deep lakes [15].
- Empirical equation methods: These methods establish a formula for estimating the average depth with the lake area based on a large number of real average-depth data to calibrate the parameters [16]. Some other method even takes more information into account such as surrounding average slope [1]. However, they cannot generate bathymetric maps or derive the volume-depth and area-depth curves for single lakes.
- Similar volume–area (V–A) curve method: This method is based on the regional similarity of volume–area relationships. The method chooses several similar valleys as virtual reservoirs and then simulates the process of water filling and regression. It may be successful for researching small mountainous lakes, but for medium and large lakes, similar valleys are always lacking [17].
- Lake bathymetric map simulation method: This method uses the surrounding digital elevation model (DEM) data to estimate the lake bottom shape. For example, Messager et al. tested a GIS-based method based on power functions in which the depth is derived as ((the distance from shore)^α × tan(slope)) using different values of the exponent α [1]. We reproduced this method and found that the model result cannot maintain continuity, which is the most challenging barrier in the model proposed in this study. We finally overcame this problem by proposing a water recession method (WRM).
- The area–elevation combined methods: These methods use optical satellite imagery such as Landsat [18] or MODIS [19,20] to derive lake-surface area series; meanwhile, they use satellite altimetry such as ICESat/GLAS and ENVISAT [18,19] to generate surface elevation. The area and surface elevation series are then combined to calculate relative lake volume estimates. These methods are widely acceptable since they are based on simple and direct mathematical logics. However, the lack and disconnection of lake level data from satellite altimetry is a key issue that limits the applications of the abovementioned methods [18,21]. Besides, these methods can achieve accurate results of relative lake volume rather than absolute volume or bathymetry.
- Other interpolating methods and derivative spline methods: These methods basically use the control points in the calculation area and the control conditions of boundaries to generate the whole bathymetric map for water bodies, such as Kriging [22] or spline interpolation [23]. There are many successful researches about rivers because of the regularity of the river bathymetry [24,25], and there are even some mature methods in ArcGIS tools [26], but the common point is that they need field survey results as input data. We admit the measured data is indispensable for high accurate research, and more uniform distribution of the input data will lead to more accurate results. However, in considering the big cost of large-scale in-situ data [5], it is still important to develop remote sensing methods without these data. This is also the starting point of our method.

## 2. Data and Model Development

#### 2.1. Input Data

#### 2.2. Assumptions

- The natural surface of the lake’s bathymetry is typically formed and shaped by geophysical processes similar to those that shaped its surrounding landmass.
- The agent rate of water is uniform throughout the whole lake.

#### 2.3. Model Procedure

- Finding CCPs: The current calculation points (CCPs) are found through a circulation step by (1) drawing the AGA border adjacent to the uncalculated side, (2) finding the pixels whose values are higher than the H, and (3) defining the pixels that are in the uncalculated area but next to the pixels found in step 2 as the CCPs. For example, the hollow red point in Figure 3 is a CCP ready for calculation. Note that the CCP is the pixel that is calculated during each circulation interval.
- Determining BOD of CCPs: After the CCPs are found, we need to calculate each CCP based on the same AGA. For one CCP, before we apply the calculation formula, we need to distinguish the border-on directions (BODs), defined as the directions on which we can find a border-on AGA pixel. There are four potential directions (north, west, south, and east) for each CCP, and the BODs are defined as the directions on which we can find a border-on AGA pixel. As shown in Figure 3, the north and west directions are the BODs for this CCP. The reason for finding the BODs is based on the inverse distance weighted interpolation method [39], meaning that the influencing power decreases as the distance increases. In this model, we consider that the far coast has little influence on nearshore pixels, so the pixel elevation is only decided from the BODs. Obviously, the last pixel should have four BODs, while most cases have one to three BODs.
- Estimating the depth value of CCPs by the MFM: After the BODs are determined, we use the morphologic function module (MFM) to calculate the estimated pixel elevation from all BODs and then average the results (please see Section 2.5 for more details). This step iterates until the following situation is met.
- After obtaining the value of the CCPs, by now, all CCPs are assigned an estimated value in this iteration, the AGA and H will be updated to the next iteration. For the AGA, the position of the CCPs in this iteration will turn to “calculated” from “uncalculated”. As for H, after step 3, if all elevations of the AGA boundary drop below H, we then need to decrease H to make the calculation continue by a predefined decreasing step parameter (DSP); if not, it will do another iteration using the same H. In the former case, the new H equals the original H minus the DSP, and this process is continued until any CCP occurs. The DSP has a low sensitivity when it is small enough, but it also influences the efficiency of the program if it is too small. In this study, we set the DSP as 1 m in all tests with the same vertical resolution as the SRTM data. The H and DSP changing mechanism are the core of the WRM, and it makes the calculation progress relying on the water surface decrease rather than simply shrink the geometry boundary; this is better since the WRM imitates a natural process that makes the calculated surface almost horizontal rather than irregular.

#### 2.4. Surrounding Slope Module (SSM)

#### 2.5. Morphologic Function Module (MFM)

## 3. Model Validation

#### 3.1. Analysis of Lake Ontario

#### 3.2. Analysis of Lake Namco

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**The common process of how a lake is formed. The black line represents the natural surface; the yellow line/points represent the sediment surface/material; and the blue line represents the water surface.

## Appendix B

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**Figure 1.**The deposition cross-section of the lake profile. The blue line represents the lake water surface, the black line represents the natural bottom of the lake, and the yellow area represents the sediment layer. The Sc is a scale value, equal to the water depth divided by the distance between the lake’s water surface and the natural bottom. The Sc becomes 1 when there is no sediment.

**Figure 3.**Model calculation procedure diagram. The black line represents the original bank of the lake, and the green line represents the calculated bank of the lake. The red circle is the current calculated pixel, and the red/blue solid dots represent the linked original/current boundary points. S and s

_{i}represent the original and current calculation widths, respectively, which can be shown in the profile in Figure 5.

**Figure 4.**Slope calculation diagram, with land and lake pixels marked as black and blue, respectively. The red point is the slope calculation point, whose location is shown in a plan view in Figure 3. The $P\left(i\right)$ is the slope from the land to lake as indicated by the red arrow symbol, with the black arrow symbol on the bottom showing the direction of calculation. $k1$ is the final average slope value, which is calculated from Equation (2).

**Figure 5.**The calculation diagram of the MFM (morphologic function module). The black line represents the estimated lake bottom. The red circle is the pixel which is calculated now, and the red/blue solid points represent the linked original/current boundary points (same as in Figure 3). The yellow point represents the assumed bottom of the lake with zero slope. $h1$ and $h2$ are the elevations of original boundary points (actual lake surface level with the same values); $h{1}_{i}$ and $h{2}_{i}$ are the elevations at current boundary points, meanwhile they are both assumed to be equal to ${h}_{i}$ due to small $k{0}_{i}$ values. $k1$ and $k2$ are slope data with the corresponding direction from the SSD (Figure 4); and similarly, $k{1}_{i}$ and $k{2}_{i}$ are the slopes at current boundary points through calculation. Other parameters: $S$ and ${s}_{i}$ represent the original/current calculation width, respectively; $h0$ and ${h}_{i}$ represent the original/current calculation depths, respectively; $D1$ and $D2$ represent the horizontal distance between the assumed bottom and original boundary points; similarly, $d{1}_{i}$ and $d{2}_{i}$ represent the horizontal distance between the assumed bottom and current boundary points. Furthermore, the $k{0}_{i}$ is a small decimal deviation because the data is recorded by double float format.

**Figure 6.**The modeling results of Lake Ontario. (

**a**) 3D map of the measured bathymetric data; (

**b**) contour map of the measured bathymetric data; (

**c**) 3D map of the result using scaled model 1; (

**d**) contour map of the result using scaled model 1; (

**e**) 3D map of the result using scaled model 2; (

**f**) contour map of the result using scaled model 2.

**Figure 7.**The (

**a**) area-depth curve and (

**b**) volume-depth curve of Lake Ontario in model 1 and model 2.

**Figure 8.**The modeling results of Lake Namco. (

**a**) 3D map of the result using scaled model 1; (

**b**) contour map of the result using scaled model 1.

**Figure 9.**The (

**a**) lake level from Hydroweb, (

**b**) area, and (

**c**) relative volume series result of the scaled model and reference data (from Hydroweb) from 2000 to 2017.

**Figure 10.**The fitting result of the (

**a**) area and (

**b**) volume series between the Hydroweb data and the proposed model.

Abbreviation | Full name | Definition |
---|---|---|

WRM | Water Recession Method | It is a process simulating the water recession. In this process, the water keeps decreasing and land appears gradually. |

SSM | Surrounding Slope Module | It is a program module designed for calculating the surrounding slope of four directions |

SSD | Surrounding Slope Data | It is the result of the SSM recorded by a $\mathrm{m}\times \mathrm{n}\times 4$ matrix. The value is slope, m and n represent position, and 4 represents direction. |

LBM | Lake Binary Matrix | It is a binary matrix to record water and land by 1 and 0, and it has the same size with input from the digital elevation model (DEM). |

H | Height | It is the current calculation of elevation, as well as the assumed water surface height in the WRM. |

AGA | Already Generated Area | It is a binary matrix to record the calculated and uncalculated pixels, and it has the same format as that of the LBM. |

DSP | Decreasing Step Parameter | It is the decreasing height of every single iteration and it controls the decreasing speed of the water surface in the WRM. |

CCP | Current Calculation Point | The pixel which will be assigned to an estimated value in this present iteration. |

BOD | Border-On Direction | In this direction, the CCP is adjacent to the water pixel rather than the land pixel. |

MFM | Morphologic Function Module | It is a program module designed to calculate the estimated value of the CCP by mathematical function and input information. |

Sc | Sediment coefficient | It is a correction factor serving for this model based on the two assumptions. |

Model | MAE | RMSE | R-squared of A ^{1} | R-squared of V ^{1} |
---|---|---|---|---|

1 | 24.232 m | 33.957 m | 0.982 | 0.993 |

2 | 22.076 m | 29.312 m | 0.973 | 0.988 |

^{1}R-squared of area/volume (A/V) represents the R-squared calculated by area/volume series data. MAE, mean absolute error; RMSE, root mean square error.

Model | Area | Volume |
---|---|---|

Unscaled | 53.669 ${\mathrm{km}}^{2}$ | 0.123 ${\mathrm{km}}^{3}$ |

Scaled | 55.896 ${\mathrm{km}}^{2}$ | 0.135 ${\mathrm{km}}^{3}$ |

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## Share and Cite

**MDPI and ACS Style**

Zhu, S.; Liu, B.; Wan, W.; Xie, H.; Fang, Y.; Chen, X.; Li, H.; Fang, W.; Zhang, G.; Tao, M.;
et al. A New Digital Lake Bathymetry Model Using the Step-Wise Water Recession Method to Generate 3D Lake Bathymetric Maps Based on DEMs. *Water* **2019**, *11*, 1151.
https://doi.org/10.3390/w11061151

**AMA Style**

Zhu S, Liu B, Wan W, Xie H, Fang Y, Chen X, Li H, Fang W, Zhang G, Tao M,
et al. A New Digital Lake Bathymetry Model Using the Step-Wise Water Recession Method to Generate 3D Lake Bathymetric Maps Based on DEMs. *Water*. 2019; 11(6):1151.
https://doi.org/10.3390/w11061151

**Chicago/Turabian Style**

Zhu, Siyu, Baojian Liu, Wei Wan, Hongjie Xie, Yu Fang, Xi Chen, Huan Li, Weizhen Fang, Guoqing Zhang, Mingwei Tao,
and et al. 2019. "A New Digital Lake Bathymetry Model Using the Step-Wise Water Recession Method to Generate 3D Lake Bathymetric Maps Based on DEMs" *Water* 11, no. 6: 1151.
https://doi.org/10.3390/w11061151