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Impact of terrain complexity on the accuracy of calculations of river channel storage volume derived from measurements of underwater topography

  • Original Paper
  • Published:
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Abstract

In this paper, we discuss the approximate errors that accrue in the approximation of continuous riverbed surfaces using data from discrete topographic surveying points, by evaluating the accuracy of the river channel storage volume calculated from underwater topographic surveying data without acknowledging the measurement errors inherent in data acquisition. Two types of theoretical surfaces with different complexity levels were generated from mathematical expressions and fractal Brownian motion. The reference value of the volume between a theoretical surface and a horizontal reference plane was calculated. Regularly distributed elevation spots were sampled from the theoretical surface to match those sampled in the topographic surveying of the river channel. The volume corresponding to the scale of terrain survey and the relative error of the reference value were computed using the scattered data points. The complexity of the theoretical surface was characterized and quantified using the fractal dimension. Regression analysis showed that there was an apparent positive correlation between the fractal dimension and the relative error, and the correlation coefficient (R 2) was greater than 0.794 for the three case studies of smooth surfaces and nearly 1.0 for the two case studies of rough surfaces. It indicates that the data representativeness of scattered points with the same point density in a flat surface is better than that in a complicated surface. From this study, we can conclude that the topographic fluctuation of the riverbed has an effect on the calculation error of the river channel storage volume derived from measurements of underwater topography, and its degree of influence depends on the complexity level of the terrain. Moreover, this paper provide a distinct approach for analyzing the measurement representation in a terrain survey and more work will involve surface modeling techniques combined with realistic topographic feature to produce more vivid landform.

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Acknowledgments

The China National Natural Science Foundation (Grant No. 51379155) supported this work.

Author information

Authors and Affiliations

  1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, China

    Xiaofeng Zhang, Yuan Yuan & Mengji Qi

Authors
  1. Xiaofeng Zhang
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  2. Yuan Yuan
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  3. Mengji Qi
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Corresponding author

Correspondence to Yuan Yuan.

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Appendix

Appendix

  • Surface 3–1:

    $$ z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{100}\cdot x\right)+ \cos \left(\frac{\pi }{100}\cdot x\right)+2 \sin \left(\frac{2\pi }{100}\cdot x\right)+2 \cos \left(\frac{2\pi }{100}\cdot x\right),0\le x<200;\hfill \\ {} \sin \left[\frac{\pi }{105}\cdot \left(x-200\right)\right]+ \cos \left[\frac{\pi }{105}\cdot \left(x-200\right)\right]+2 \sin \left[\frac{2\pi }{105}\cdot \left(x-200\right)\right]+2 \cos \left[\frac{2\pi }{105}\cdot \left(x-200\right)\right],200\le x<410;\hfill \\ {} \sin \left[\frac{\pi }{110}\cdot \left(x-410\right)\right]+ \cos \left[\frac{\pi }{110}\cdot \left(x-410\right)\right]+2 \sin \left[\frac{2\pi }{110}\cdot \left(x-410\right)\right]+2 \cos \left[\frac{2\pi }{110}\cdot \left(x-410\right)\right],410\le x<630;\hfill \\ {} \sin \left[\frac{\pi }{115}\cdot \left(x-630\right)\right]+ \cos \left[\frac{\pi }{115}\cdot \left(x-630\right)\right]+2 \sin \left[\frac{2\pi }{115}\cdot \left(x-630\right)\right]+2 \cos \left[\frac{2\pi }{115}\cdot \left(x-630\right)\right],630\le x<860;\hfill \\ {} \sin \left[\frac{\pi }{120}\cdot \left(x-860\right)\right]+ \cos \left[\frac{\pi }{120}\cdot \left(x-860\right)\right]+2 \sin \left[\frac{2\pi }{120}\cdot \left(x-860\right)\right]+2 \cos \left[\frac{2\pi }{120}\cdot \left(x-860\right)\right],860\le x<1100;\hfill \\ {} \sin \left[\frac{\pi }{125}\cdot \left(x-1100\right)\right]+ \cos \left[\frac{\pi }{125}\cdot \left(x-1100\right)\right]+2 \sin \left[\frac{2\pi }{125}\cdot \left(x-1100\right)\right]+2 \cos \left[\frac{2\pi }{125}\cdot \left(x-1100\right)\right],1100\le x<1350;\hfill \\ {} \sin \left[\frac{\pi }{130}\cdot \left(x-1350\right)\right]+ \cos \left[\frac{\pi }{130}\cdot \left(x-1350\right)\right]+2 \sin \left[\frac{2\pi }{130}\cdot \left(x-1350\right)\right]+2 \cos \left[\frac{2\pi }{130}\cdot \left(x-1350\right)\right],1350\le x<1610;\hfill \\ {} \sin \left[\frac{\pi }{135}\cdot \left(x-1610\right)\right]+ \cos \left[\frac{\pi }{135}\cdot \left(x-1610\right)\right]+2 \sin \left[\frac{2\pi }{135}\cdot \left(x-1610\right)\right]+2 \cos \left[\frac{2\pi }{135}\cdot \left(x-1610\right)\right],1610\le x<1880;\hfill \\ {} \sin \left[\frac{\pi }{60}\cdot \left(x-1880\right)\right]+ \cos \left[\frac{\pi }{60}\cdot \left(x-1880\right)\right]+2 \sin \left[\frac{2\pi }{60}\cdot \left(x-1880\right)\right]+2 \cos \left[\frac{2\pi }{60}\cdot \left(x-1880\right)\right],1880\le x<2000;\hfill \end{array}\right. $$
    (11)

  • Surface 3–2:

    $$ z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{105}\cdot x\right)+ \cos \left(\frac{\pi }{105}\cdot x\right)+2 \sin \left(\frac{2\pi }{105}\cdot x\right)+2 \cos \left(\frac{2\pi }{105}\cdot x\right),0\le x<210;\hfill \\ {} \sin \left[\frac{\pi }{115}\cdot \left(x-210\right)\right]+ \cos \left[\frac{\pi }{115}\cdot \left(x-210\right)\right]+2 \sin \left[\frac{2\pi }{115}\cdot \left(x-210\right)\right]+2 \cos \left[\frac{2\pi }{115}\cdot \left(x-210\right)\right],210\le x<440;\hfill \\ {} \sin \left[\frac{\pi }{130}\cdot \left(x-440\right)\right]+ \cos \left[\frac{\pi }{130}\cdot \left(x-440\right)\right]+2 \sin \left[\frac{2\pi }{130}\cdot \left(x-440\right)\right]+2 \cos \left[\frac{2\pi }{130}\cdot \left(x-440\right)\right],440\le x<700;\hfill \\ {} \sin \left[\frac{\pi }{145}\cdot \left(x-700\right)\right]+ \cos \left[\frac{\pi }{145}\cdot \left(x-700\right)\right]+2 \sin \left[\frac{2\pi }{145}\cdot \left(x-700\right)\right]+2 \cos \left[\frac{2\pi }{145}\cdot \left(x-700\right)\right],700\le x<990;\hfill \\ {} \sin \left[\frac{\pi }{160}\cdot \left(x-990\right)\right]+ \cos \left[\frac{\pi }{160}\cdot \left(x-990\right)\right]+2 \sin \left[\frac{2\pi }{160}\cdot \left(x-990\right)\right]+2 \cos \left[\frac{2\pi }{160}\cdot \left(x-990\right)\right],990\le x<1310;\hfill \\ {} \sin \left[\frac{\pi }{170}\cdot \left(x-1310\right)\right]+ \cos \left[\frac{\pi }{170}\cdot \left(x-1310\right)\right]+2 \sin \left[\frac{2\pi }{170}\cdot \left(x-1310\right)\right]+2 \cos \left[\frac{2\pi }{170}\cdot \left(x-1310\right)\right],1310\le x<1650;\hfill \\ {} \sin \left[\frac{\pi }{175}\cdot \left(x-1650\right)\right]+ \cos \left[\frac{\pi }{175}\cdot \left(x-1650\right)\right]+2 \sin \left[\frac{2\pi }{175}\cdot \left(x-1650\right)\right]+2 \cos \left[\frac{2\pi }{175}\cdot \left(x-1650\right)\right],1650\le x<2000;\hfill \end{array}\right. $$
    (12)

  • Surface 3–3:

    $$ z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{110}\cdot x\right)+ \cos \left(\frac{\pi }{110}\cdot x\right)+2 \sin \left(\frac{2\pi }{110}\cdot x\right)+2 \cos \left(\frac{2\pi }{110}\cdot x\right),0\le x<220;\hfill \\ {} \sin \left[\frac{\pi }{130}\cdot \left(x-220\right)\right]+ \cos \left[\frac{\pi }{130}\cdot \left(x-220\right)\right]+2 \sin \left[\frac{2\pi }{130}\cdot \left(x-220\right)\right]+2 \cos \left[\frac{2\pi }{130}\cdot \left(x-220\right)\right],220\le x<480;\hfill \\ {} \sin \left[\frac{\pi }{150}\cdot \left(x-480\right)\right]+ \cos \left[\frac{\pi }{150}\cdot \left(x-480\right)\right]+2 \sin \left[\frac{2\pi }{150}\cdot \left(x-480\right)\right]+2 \cos \left[\frac{2\pi }{150}\cdot \left(x-480\right)\right],480\le x<780;\hfill \\ {} \sin \left[\frac{\pi }{175}\cdot \left(x-780\right)\right]+ \cos \left[\frac{\pi }{175}\cdot \left(x-780\right)\right]+2 \sin \left[\frac{2\pi }{175}\cdot \left(x-780\right)\right]+2 \cos \left[\frac{2\pi }{175}\cdot \left(x-780\right)\right],780\le x<1130;\hfill \\ {} \sin \left[\frac{\pi }{200}\cdot \left(x-1130\right)\right]+ \cos \left[\frac{\pi }{200}\cdot \left(x-1130\right)\right]+2 \sin \left[\frac{2\pi }{200}\cdot \left(x-1130\right)\right]+2 \cos \left[\frac{2\pi }{200}\cdot \left(x-1130\right)\right],1130\le x<1530;\hfill \\ {} \sin \left[\frac{\pi }{235}\cdot \left(x-1530\right)\right]+ \cos \left[\frac{\pi }{235}\cdot \left(x-1530\right)\right]+2 \sin \left[\frac{2\pi }{235}\cdot \left(x-1530\right)\right]+2 \cos \left[\frac{2\pi }{235}\cdot \left(x-1530\right)\right],1530\le x<2000;\hfill \end{array}\right. $$
    (13)

  • Surface 3–4:

    $$ \begin{array}{l}z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{115}\cdot x\right)+ \cos \left(\frac{\pi }{115}\cdot x\right)+2 \sin \left(\frac{2\pi }{115}\cdot x\right)+2 \cos \left(\frac{2\pi }{115}\cdot x\right),0\le x<230;\\ {} \sin \left[\frac{\pi }{135}\cdot \left(x-230\right)\right]+ \cos \left[\frac{\pi }{135}\cdot \left(x-230\right)\right]+2 \sin \left[\frac{2\pi }{135}\cdot \left(x-230\right)\right]+2 \cos \left[\frac{2\pi }{135}\cdot \left(x-230\right)\right],230\le x<500;\\ {} \sin \left[\frac{\pi }{155}\cdot \left(x-500\right)\right]+ \cos \left[\frac{\pi }{155}\cdot \left(x-500\right)\right]+2 \sin \left[\frac{2\pi }{155}\cdot \left(x-500\right)\right]+2 \cos \left[\frac{2\pi }{155}\cdot \left(x-500\right)\right],500\le x<810;\\ {} \sin \left[\frac{\pi }{180}\cdot \left(x-810\right)\right]+ \cos \left[\frac{\pi }{180}\cdot \left(x-810\right)\right]+2 \sin \left[\frac{2\pi }{180}\cdot \left(x-810\right)\right]+2 \cos \left[\frac{2\pi }{180}\cdot \left(x-810\right)\right],810\le x<1170;\\ {} \sin \left[\frac{\pi }{155}\cdot \left(x-1170\right)\right]+ \cos \left[\frac{\pi }{155}\cdot \left(x-1170\right)\right]+2 \sin \left[\frac{2\pi }{155}\cdot \left(x-1170\right)\right]+2 \cos \left[\frac{2\pi }{155}\cdot \left(x-1170\right)\right],1170\le x<1480;\\ {} \sin \left[\frac{\pi }{135}\cdot \left(x-1480\right)\right]+ \cos \left[\frac{\pi }{135}\cdot \left(x-1480\right)\right]+2 \sin \left[\frac{2\pi }{135}\cdot \left(x-1480\right)\right]+2 \cos \left[\frac{2\pi }{135}\cdot \left(x-1480\right)\right],1480\le x<1750;\\ {} \sin \left[\frac{\pi }{125}\cdot \left(x-1750\right)\right]+ \cos \left[\frac{\pi }{125}\cdot \left(x-1750\right)\right]+2 \sin \left[\frac{2\pi }{125}\cdot \left(x-1750\right)\right]+2 \cos \left[\frac{2\pi }{125}\cdot \left(x-1750\right)\right],1750\le x<2000;\end{array}\right.\\ {}\end{array} $$
    (14)

  • Surface 3–5:

    $$ z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{160}\cdot x\right)+ \cos \left(\frac{\pi }{160}\cdot x\right)+2 \sin \left(\frac{2\pi }{160}\cdot x\right)+2 \cos \left(\frac{2\pi }{160}\cdot x\right),0\le x<320;\hfill \\ {} \sin \left[\frac{\pi }{180}\cdot \left(x-320\right)\right]+ \cos \left[\frac{\pi }{180}\cdot \left(x-320\right)\right]+2 \sin \left[\frac{2\pi }{180}\cdot \left(x-320\right)\right]+2 \cos \left[\frac{2\pi }{180}\cdot \left(x-320\right)\right],320\le x<680;\hfill \\ {} \sin \left[\frac{\pi }{200}\cdot \left(x-680\right)\right]+ \cos \left[\frac{\pi }{200}\cdot \left(x-680\right)\right]+2 \sin \left[\frac{2\pi }{200}\cdot \left(x-680\right)\right]+2 \cos \left[\frac{2\pi }{200}\cdot \left(x-680\right)\right],680\le x<1080;\hfill \\ {} \sin \left[\frac{\pi }{220}\cdot \left(x-1080\right)\right]+ \cos \left[\frac{\pi }{220}\cdot \left(x-1080\right)\right]+2 \sin \left[\frac{2\pi }{220}\cdot \left(x-1080\right)\right]+2 \cos \left[\frac{2\pi }{220}\cdot \left(x-1080\right)\right],1080\le x<1520;\hfill \\ {} \sin \left[\frac{\pi }{240}\cdot \left(x-1520\right)\right]+ \cos \left[\frac{\pi }{240}\cdot \left(x-1520\right)\right]+2 \sin \left[\frac{2\pi }{240}\cdot \left(x-1520\right)\right]+2 \cos \left[\frac{2\pi }{240}\cdot \left(x-1520\right)\right],1520\le x<2000;\hfill \end{array}\right. $$
    (15)

  • Surface 3–6:

    $$ z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{220}\cdot x\right)+ \cos \left(\frac{\pi }{220}\cdot x\right)+2 \sin \left(\frac{2\pi }{220}\cdot x\right)+2 \cos \left(\frac{2\pi }{220}\cdot x\right),0\le x<440;\hfill \\ {} \sin \left[\frac{\pi }{240}\cdot \left(x-440\right)\right]+ \cos \left[\frac{\pi }{240}\cdot \left(x-440\right)\right]+2 \sin \left[\frac{2\pi }{240}\cdot \left(x-440\right)\right]+2 \cos \left[\frac{2\pi }{240}\cdot \left(x-440\right)\right],440\le x<920;\hfill \\ {} \sin \left[\frac{\pi }{260}\cdot \left(x-920\right)\right]+ \cos \left[\frac{\pi }{260}\cdot \left(x-920\right)\right]+2 \sin \left[\frac{2\pi }{260}\cdot \left(x-920\right)\right]+2 \cos \left[\frac{2\pi }{260}\cdot \left(x-920\right)\right],920\le x<1440;\hfill \\ {} \sin \left[\frac{\pi }{280}\cdot \left(x-1440\right)\right]+ \cos \left[\frac{\pi }{280}\cdot \left(x-1440\right)\right]+2 \sin \left[\frac{2\pi }{280}\cdot \left(x-1440\right)\right]+2 \cos \left[\frac{2\pi }{280}\cdot \left(x-1440\right)\right],1440\le x<2000;\hfill \end{array}\right. $$
    (16)

  • Surface 3–7:

    $$ z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{275}\cdot x\right)+ \cos \left(\frac{\pi }{275}\cdot x\right)+2 \sin \left(\frac{2\pi }{275}\cdot x\right)+2 \cos \left(\frac{2\pi }{275}\cdot x\right),0\le x<550;\hfill \\ {} \sin \left[\frac{\pi }{325}\cdot \left(x-550\right)\right]+ \cos \left[\frac{\pi }{325}\cdot \left(x-550\right)\right]+2 \sin \left[\frac{2\pi }{325}\cdot \left(x-550\right)\right]+2 \cos \left[\frac{2\pi }{325}\cdot \left(x-550\right)\right],550\le x<1200;\hfill \\ {} \sin \left[\frac{\pi }{400}\cdot \left(x-1200\right)\right]+ \cos \left[\frac{\pi }{400}\cdot \left(x-1200\right)\right]+2 \sin \left[\frac{2\pi }{400}\cdot \left(x-1200\right)\right]+2 \cos \left[\frac{2\pi }{400}\cdot \left(x-1200\right)\right],1200\le x<2000;\hfill \end{array}\right. $$
    (17)

  • Surface 3–8:

    $$ z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{105}\cdot x\right)+ \cos \left(\frac{\pi }{105}\cdot x\right)+2 \sin \left(\frac{2\pi }{105}\cdot x\right)+2 \cos \left(\frac{2\pi }{105}\cdot x\right),0\le x<210;\hfill \\ {} \sin \left[\frac{\pi }{120}\cdot \left(x-210\right)\right]+ \cos \left[\frac{\pi }{120}\cdot \left(x-210\right)\right]+2 \sin \left[\frac{2\pi }{120}\cdot \left(x-210\right)\right]+2 \cos \left[\frac{2\pi }{120}\cdot \left(x-210\right)\right],210\le x<450;\hfill \\ {} \sin \left[\frac{\pi }{130}\cdot \left(x-450\right)\right]+ \cos \left[\frac{\pi }{130}\cdot \left(x-450\right)\right]+2 \sin \left[\frac{2\pi }{130}\cdot \left(x-450\right)\right]+2 \cos \left[\frac{2\pi }{130}\cdot \left(x-450\right)\right],450\le x<710;\hfill \\ {} \sin \left[\frac{\pi }{140}\cdot \left(x-710\right)\right]+ \cos \left[\frac{\pi }{140}\cdot \left(x-710\right)\right]+2 \sin \left[\frac{2\pi }{140}\cdot \left(x-710\right)\right]+2 \cos \left[\frac{2\pi }{140}\cdot \left(x-710\right)\right],710\le x<990;\hfill \\ {} \sin \left[\frac{\pi }{135}\cdot \left(x-990\right)\right]+ \cos \left[\frac{\pi }{135}\cdot \left(x-990\right)\right]+2 \sin \left[\frac{2\pi }{135}\cdot \left(x-990\right)\right]+2 \cos \left[\frac{2\pi }{135}\cdot \left(x-990\right)\right],990\le x<1260;\hfill \\ {} \sin \left[\frac{\pi }{125}\cdot \left(x-1260\right)\right]+ \cos \left[\frac{\pi }{125}\cdot \left(x-1260\right)\right]+2 \sin \left[\frac{2\pi }{125}\cdot \left(x-1260\right)\right]+2 \cos \left[\frac{2\pi }{125}\cdot \left(x-1260\right)\right],1260\le x<1510;\hfill \\ {} \sin \left[\frac{\pi }{115}\cdot \left(x-1510\right)\right]+ \cos \left[\frac{\pi }{115}\cdot \left(x-1510\right)\right]+2 \sin \left[\frac{2\pi }{115}\cdot \left(x-1510\right)\right]+2 \cos \left[\frac{2\pi }{115}\cdot \left(x-1510\right)\right],1510\le x<1740;\hfill \\ {} \sin \left[\frac{\pi }{130}\cdot \left(x-1740\right)\right]+ \cos \left[\frac{\pi }{130}\cdot \left(x-1740\right)\right]+2 \sin \left[\frac{2\pi }{130}\cdot \left(x-1740\right)\right]+2 \cos \left[\frac{2\pi }{130}\cdot \left(x-1740\right)\right],1740\le x<2000\hfill \end{array}\right. $$
    (18)

  • Surface 3–9:

    $$ z(m)=19+ \cos \left(\frac{\pi }{400}\cdot y\right)+\left\{\begin{array}{l} \sin \left(\frac{\pi }{120}\cdot x\right)+ \cos \left(\frac{\pi }{120}\cdot x\right)+2 \sin \left(\frac{2\pi }{120}\cdot x\right)+2 \cos \left(\frac{2\pi }{120}\cdot x\right),0\le x<240;\hfill \\ {} \sin \left[\frac{\pi }{140}\cdot \left(x-240\right)\right]+ \cos \left[\frac{\pi }{140}\cdot \left(x-240\right)\right]+2 \sin \left[\frac{2\pi }{140}\cdot \left(x-240\right)\right]+2 \cos \left[\frac{2\pi }{140}\cdot \left(x-240\right)\right],240\le x<520;\hfill \\ {} \sin \left[\frac{\pi }{160}\cdot \left(x-520\right)\right]+ \cos \left[\frac{\pi }{160}\cdot \left(x-520\right)\right]+2 \sin \left[\frac{2\pi }{160}\cdot \left(x-520\right)\right]+2 \cos \left[\frac{2\pi }{160}\cdot \left(x-520\right)\right],520\le x<840;\hfill \\ {} \sin \left[\frac{\pi }{180}\cdot \left(x-840\right)\right]+ \cos \left[\frac{\pi }{180}\cdot \left(x-840\right)\right]+2 \sin \left[\frac{2\pi }{180}\cdot \left(x-840\right)\right]+2 \cos \left[\frac{2\pi }{180}\cdot \left(x-840\right)\right],840\le x<1200;\hfill \\ {} \sin \left[\frac{\pi }{190}\cdot \left(x-1200\right)\right]+ \cos \left[\frac{\pi }{190}\cdot \left(x-1200\right)\right]+2 \sin \left[\frac{2\pi }{190}\cdot \left(x-1200\right)\right]+2 \cos \left[\frac{2\pi }{190}\cdot \left(x-1200\right)\right],1200\le x<1580;\hfill \\ {} \sin \left[\frac{\pi }{210}\cdot \left(x-1580\right)\right]+ \cos \left[\frac{\pi }{210}\cdot \left(x-1580\right)\right]+2 \sin \left[\frac{2\pi }{210}\cdot \left(x-1580\right)\right]+2 \cos \left[\frac{2\pi }{210}\cdot \left(x-1580\right)\right],1580\le x<2000\hfill \end{array}\right. $$
    (19)

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Zhang, X., Yuan, Y. & Qi, M. Impact of terrain complexity on the accuracy of calculations of river channel storage volume derived from measurements of underwater topography. Arab J Geosci 8, 9149–9168 (2015). https://doi.org/10.1007/s12517-015-1857-9

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