Hypsographic Curve Integration

Integrate observed hypsographic area-depth contours into a cumulative volume curve with trapezoids; contour geometry is supplied and only the measured depth range is represented.

Description

A hypsographic curve records lake area by depth.

A hypsographic curve records lake area by depth. Integrating that area produces cumulative water volume and mean depth over the measured depth range.^[1]

The tool returns cumulativeVolumesAboveDepthM3, totalModeledVolumeM3, meanDepthM. The computation uses supplied observations and parameters rather than acquiring measurements.

When to use Hypsographic Curve Integration

Use this for the following task: Integrate observed hypsographic area-depth contours into a cumulative volume curve with trapezoids; contour geometry is supplied and only the measured depth range is represented. Prepare observations and calibration parameters on the same unit and reporting basis.

How to use Hypsographic Curve Integration and understand its results

Prepare the measured quantities and supplied model parameters. Match the units and align paired lists before calculating.
Enter depthsM, areasM2 according to the field table, then run the tool.
Read cumulativeVolumesAboveDepthM3, totalModeledVolumeM3, meanDepthM. Use the calculation below to interpret the selected mean or the model’s signs, units and comparison flags.
Review the supplied assumptions and calibration before comparing or reporting the result.

Input field	Meaning	Value type and bounds	Requirement
depthsM	Depths m	list of number	Required
areasM2	Areas m2	list of number	Required

Provide aligned increasing depths starting at zero and nonnegative nonincreasing contour areas with positive surface area. Sum trapezoids and report cumulative volumes above each sampled depth.

interval volume=Δdepth×(areaLower+areaUpper)/2; meanDepth=totalVolume/surfaceArea

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
cumulativeVolumesAboveDepthM3	Cumulative volumes above depth m3	list of number
totalModeledVolumeM3	Total modeled volume m3	number
meanDepthM	Mean depth m	number

json

{
  "input": {
    "depthsM": [
      0,
      10,
      20
    ],
    "areasM2": [
      100,
      50,
      0
    ]
  },
  "output": {
    "cumulativeVolumesAboveDepthM3": [
      0,
      750,
      1000
    ],
    "totalModeledVolumeM3": 1000,
    "meanDepthM": 10
  }
}

Limitations and assumptions

Contours and bathymetry are supplied. No shape outside the measured range is invented; linear interpolation between contours defines the modeled volume.

Numerical inputs must be finite. Input-shape and calculation checks do not establish measurement quality or the validity of a supplied calibration.

Alternative or Complementary methods

Choose Epilimnion Hypolimnion Volume Split when you need its operation: Integrate a supplied piecewise-linear area-depth curve across thermal-layer boundaries including split contour intervals; geometry is supplied, not reconstructed from map coordinates. Choose Stage Storage Volume Interpolation for the complementary operation: Interpolate storage for measured monotone stage/storage calibration within its observed bounds; no extrapolation, contour integration or invented lake geometry.

Background reference: Digital Library of Mathematical Functions: Quadrature.

References

Digital Library of Mathematical Functions: Quadrature — National Institute of Standards and Technology

Similar or alternative tools

Epilimnion Hypolimnion Volume Split
Integrate a supplied piecewise-linear area-depth curve across thermal-layer boundaries including split contour intervals; geometry is supplied, not reconstructed from map coordinates.
Stage Storage Volume Interpolation
Interpolate storage for measured monotone stage/storage calibration within its observed bounds; no extrapolation, contour integration or invented lake geometry.