Volume of an IRREGULAR 3D shape
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I am a geographer/ecologist and I want to know how to accurately calculate volume of a lake or a reservoir? I am not looking for a vague estimate which is generally calculated using surface area and mean height parameters assuming the body is of a certain shape (truncated cone/triangle or circular). Since reservoirs are completely irregular in shape I am having difficulties in using the traditional volume formulae. Any help or suggestion would be greatly appreciated. Thank you.
geometry
asked Nov 22 at 13:09
Tejas
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I am a geographer/ecologist and I want to know how to accurately calculate volume of a lake or a reservoir? I am not looking for a vague estimate which is generally calculated using surface area and mean height parameters assuming the body is of a certain shape (truncated cone/triangle or circular). Since reservoirs are completely irregular in shape I am having difficulties in using the traditional volume formulae. Any help or suggestion would be greatly appreciated. Thank you.
geometry
asked Nov 22 at 13:09
Tejas
1
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am a geographer/ecologist and I want to know how to accurately calculate volume of a lake or a reservoir? I am not looking for a vague estimate which is generally calculated using surface area and mean height parameters assuming the body is of a certain shape (truncated cone/triangle or circular). Since reservoirs are completely irregular in shape I am having difficulties in using the traditional volume formulae. Any help or suggestion would be greatly appreciated. Thank you.
geometry
asked Nov 22 at 13:09
Tejas
1
I am a geographer/ecologist and I want to know how to accurately calculate volume of a lake or a reservoir? I am not looking for a vague estimate which is generally calculated using surface area and mean height parameters assuming the body is of a certain shape (truncated cone/triangle or circular). Since reservoirs are completely irregular in shape I am having difficulties in using the traditional volume formulae. Any help or suggestion would be greatly appreciated. Thank you.
geometry
geometry
asked Nov 22 at 13:09
Tejas
1
asked Nov 22 at 13:09
Tejas
1
asked Nov 22 at 13:09
Tejas
1
asked Nov 22 at 13:09
Tejas
1
asked Nov 22 at 13:09
Tejas
1
1
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add a comment |
2 Answers
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There won't be any formulas for a surface area with "completely irregular shape" . If you approximate it by a polygon (many straight sides, short enough to follow the shoreline closely) you can cut it into triangles and add their areas.
If you have an image or a map you can get as good an approximation as you like with a Monte-Carlo simulation. Choose many points on the map at random and see what fraction of them fall in the water.
I suspect good GIS software will tell you areas.
In any case you'll still need the mean height or knowledge of the shape of the bottom for the volume.
answered Nov 22 at 13:15
Ethan Bolker
40.6k545107
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You might get maps from the geographical survey that show altitude curves of the underwater terrain. If the lake is not too complicated you have the boundary $partial B_0$ of the lake at level zero, then simply closed curves $partial B_k$ $(kgeq1)$ giving the boundary of the lake if the water level would be lowered $k$ meters. Provide yourself of copies of these $partial B_k$, cut them out, and use a precise balance to find the area ${rm area}(B_k)$ of all these virtual smaller lakes $B_k$ in square centimeters of paper. Scale these values to square meters in reality. The total volume of the full lake $L$ then is approximately given by
$${rm vol}(L)approx{1over2}{rm area}(B_0)+sum_{kgeq1}1cdot{rm area}(B_k)qquad{rm cubic meters} .tag{1}$$
What $(1)$ represents is an approximation to the Lebesgue integral $int_{B_0} f(x,y)>{rm d}(x,y)$, where $f(x,y)$ denotes the depth of the lake at $(x,y)$.
answered Nov 22 at 14:12
Christian Blatter
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2 Answers
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2 Answers
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There won't be any formulas for a surface area with "completely irregular shape" . If you approximate it by a polygon (many straight sides, short enough to follow the shoreline closely) you can cut it into triangles and add their areas.
If you have an image or a map you can get as good an approximation as you like with a Monte-Carlo simulation. Choose many points on the map at random and see what fraction of them fall in the water.
I suspect good GIS software will tell you areas.
In any case you'll still need the mean height or knowledge of the shape of the bottom for the volume.
answered Nov 22 at 13:15
Ethan Bolker
40.6k545107
add a comment |
up vote
0
down vote
There won't be any formulas for a surface area with "completely irregular shape" . If you approximate it by a polygon (many straight sides, short enough to follow the shoreline closely) you can cut it into triangles and add their areas.
If you have an image or a map you can get as good an approximation as you like with a Monte-Carlo simulation. Choose many points on the map at random and see what fraction of them fall in the water.
I suspect good GIS software will tell you areas.
In any case you'll still need the mean height or knowledge of the shape of the bottom for the volume.
answered Nov 22 at 13:15
Ethan Bolker
40.6k545107
add a comment |
up vote
0
down vote
up vote
0
down vote
There won't be any formulas for a surface area with "completely irregular shape" . If you approximate it by a polygon (many straight sides, short enough to follow the shoreline closely) you can cut it into triangles and add their areas.
If you have an image or a map you can get as good an approximation as you like with a Monte-Carlo simulation. Choose many points on the map at random and see what fraction of them fall in the water.
I suspect good GIS software will tell you areas.
In any case you'll still need the mean height or knowledge of the shape of the bottom for the volume.
answered Nov 22 at 13:15
Ethan Bolker
40.6k545107
There won't be any formulas for a surface area with "completely irregular shape" . If you approximate it by a polygon (many straight sides, short enough to follow the shoreline closely) you can cut it into triangles and add their areas.
If you have an image or a map you can get as good an approximation as you like with a Monte-Carlo simulation. Choose many points on the map at random and see what fraction of them fall in the water.
I suspect good GIS software will tell you areas.
In any case you'll still need the mean height or knowledge of the shape of the bottom for the volume.
answered Nov 22 at 13:15
Ethan Bolker
40.6k545107
answered Nov 22 at 13:15
Ethan Bolker
40.6k545107
answered Nov 22 at 13:15
Ethan Bolker
40.6k545107
answered Nov 22 at 13:15
Ethan Bolker
40.6k545107
40.6k545107
add a comment |
add a comment |
up vote
0
down vote
You might get maps from the geographical survey that show altitude curves of the underwater terrain. If the lake is not too complicated you have the boundary $partial B_0$ of the lake at level zero, then simply closed curves $partial B_k$ $(kgeq1)$ giving the boundary of the lake if the water level would be lowered $k$ meters. Provide yourself of copies of these $partial B_k$, cut them out, and use a precise balance to find the area ${rm area}(B_k)$ of all these virtual smaller lakes $B_k$ in square centimeters of paper. Scale these values to square meters in reality. The total volume of the full lake $L$ then is approximately given by
$${rm vol}(L)approx{1over2}{rm area}(B_0)+sum_{kgeq1}1cdot{rm area}(B_k)qquad{rm cubic meters} .tag{1}$$
What $(1)$ represents is an approximation to the Lebesgue integral $int_{B_0} f(x,y)>{rm d}(x,y)$, where $f(x,y)$ denotes the depth of the lake at $(x,y)$.
answered Nov 22 at 14:12
Christian Blatter
171k7111325
add a comment |
up vote
0
down vote
You might get maps from the geographical survey that show altitude curves of the underwater terrain. If the lake is not too complicated you have the boundary $partial B_0$ of the lake at level zero, then simply closed curves $partial B_k$ $(kgeq1)$ giving the boundary of the lake if the water level would be lowered $k$ meters. Provide yourself of copies of these $partial B_k$, cut them out, and use a precise balance to find the area ${rm area}(B_k)$ of all these virtual smaller lakes $B_k$ in square centimeters of paper. Scale these values to square meters in reality. The total volume of the full lake $L$ then is approximately given by
$${rm vol}(L)approx{1over2}{rm area}(B_0)+sum_{kgeq1}1cdot{rm area}(B_k)qquad{rm cubic meters} .tag{1}$$
What $(1)$ represents is an approximation to the Lebesgue integral $int_{B_0} f(x,y)>{rm d}(x,y)$, where $f(x,y)$ denotes the depth of the lake at $(x,y)$.
answered Nov 22 at 14:12
Christian Blatter
171k7111325
add a comment |
up vote
0
down vote
up vote
0
down vote
You might get maps from the geographical survey that show altitude curves of the underwater terrain. If the lake is not too complicated you have the boundary $partial B_0$ of the lake at level zero, then simply closed curves $partial B_k$ $(kgeq1)$ giving the boundary of the lake if the water level would be lowered $k$ meters. Provide yourself of copies of these $partial B_k$, cut them out, and use a precise balance to find the area ${rm area}(B_k)$ of all these virtual smaller lakes $B_k$ in square centimeters of paper. Scale these values to square meters in reality. The total volume of the full lake $L$ then is approximately given by
$${rm vol}(L)approx{1over2}{rm area}(B_0)+sum_{kgeq1}1cdot{rm area}(B_k)qquad{rm cubic meters} .tag{1}$$
What $(1)$ represents is an approximation to the Lebesgue integral $int_{B_0} f(x,y)>{rm d}(x,y)$, where $f(x,y)$ denotes the depth of the lake at $(x,y)$.
answered Nov 22 at 14:12
Christian Blatter
171k7111325
You might get maps from the geographical survey that show altitude curves of the underwater terrain. If the lake is not too complicated you have the boundary $partial B_0$ of the lake at level zero, then simply closed curves $partial B_k$ $(kgeq1)$ giving the boundary of the lake if the water level would be lowered $k$ meters. Provide yourself of copies of these $partial B_k$, cut them out, and use a precise balance to find the area ${rm area}(B_k)$ of all these virtual smaller lakes $B_k$ in square centimeters of paper. Scale these values to square meters in reality. The total volume of the full lake $L$ then is approximately given by
$${rm vol}(L)approx{1over2}{rm area}(B_0)+sum_{kgeq1}1cdot{rm area}(B_k)qquad{rm cubic meters} .tag{1}$$
What $(1)$ represents is an approximation to the Lebesgue integral $int_{B_0} f(x,y)>{rm d}(x,y)$, where $f(x,y)$ denotes the depth of the lake at $(x,y)$.
answered Nov 22 at 14:12
Christian Blatter
171k7111325
answered Nov 22 at 14:12
Christian Blatter
171k7111325
answered Nov 22 at 14:12
Christian Blatter
171k7111325
answered Nov 22 at 14:12
Christian Blatter
171k7111325
171k7111325
add a comment |
add a comment |
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