Lake metabolism modeling from sensor network data

Lake metabolism modeling from sensor network dataPan-American Sensors for Environmental Observatories (PASEO), 2009, Bahia Blanca, Argentina

Paul Hanson, Tim Kratz, and Luke WinslowUniversity of Wisconsin, Center for Limnology

Support provided by

Mellon FoundationGordon & Betty Moore Foundation

“A skilled limnologist can probably learn more about the nature of a lake from a series of oxygen determinations than from any other kind of chemical data.”

G. Evelyn Hutchinson (1957)

Dissolved gases, through their observable changes through time,

allow us to understand what cannot be seen – the way lakes work.

Observable veneer

In ecosystems, the connections are not obvious or even observable, and they are physical, chemical, and biological in nature.

CB

P

CB

P

C

B

P

S

The way things work

P = physical processC = chemical processB = biological processS = state variable

Theory

Observations

Models

What can be observed?What spatio-temporal scale?Do we intervene or control?

Is it a population to be sampled?Dynamic through space and time?Are relationships empirical or mechanistic?

What are the process rates?What’s the importance to the larger story?

• Dissolved gas basics• The simple approach to calculating

lake metabolism• In truth, it’s complicated

OutlineOutline

6

7

8

9

10

11

12

237 237.5 238 238.5 239

Lake Taihu, China

Trout Bog, U.S.A.

Ormajarvi, Finland

Sparkling L., U.S.A.

Rotorua L., New Zealand

Dis

solv

ed o

xyge

n (m

g L-1

)

Day 1 Day 2

6

7

8

9

10

11

12

237 237.5 238 238.5 239

Lake Taihu, China

Trout Bog, U.S.A.

Ormajarvi, Finland

Sparkling L., U.S.A.

Rotorua L., New Zealand

Dis

solv

ed o

xyge

n (m

g L-1

)

Day 1 Day 2

GPP +RGPP +RRR

Examples of dissolved oxygen saturation over 10 days

(obtained from GLEON, using VaDER)

Lake Mendota

Sparkling Lake

Crystal Bog Lake

Date in 2008

Dis

solv

ed O

xyge

n (%

sat

)

atmosphere

water

210,000 µatm x 1.26 x10-3

370 µatm x 3.39 x10-2

= 265 µmol L-1 (~8.5 mg L-1)

= 13 µmol L-1 (~0.6 mg L-1)

O2

CO2

partial pressureHenry’s*

constant (25°C)concentration

in waterx =

Gas Partial Pressure (atm)

Nitrogen 0.78

Oxygen 0.21

Argon 0.01

Carbon dioxide 0.000370

*Henry’s constant (mol atm-1) is a function of temperature and salinity

Dissolved Gases in Fresh Water3.2 Solubility

Common units of O2 and CO2common gas pressure units:

1 atmosphere = 1013 millibars = 101 kilopascals

common dissolved gas units (concentration):O2 (DO): 1 mg L-1 x (32 mg mmol-1)-1

x 1000 µmol mmol-1 = 31.3 µmol L-1

CO2: 1 mg L-1 x (44 mg mmol-1)-1

x 1000 µmol mmol-1 = 22.7 µmol L-1

common dissolved gas units (areal): g m-2

3.1 Units of measure

Temperature (°C)

CO

2 (mg L

-1)

DO saturation

CO2 saturation

Saturation Gas Concentrations(in equilibrium with the atmosphere)

supersaturation

undersaturation

supersaturation

undersaturation

19.2

6.4

0.36

0.24

0.12

DO

(m

g L-1

)

12.8

PhotosynthesisPhotosynthesis and Respiration6CO2 + 6H2O C6H12O6 + 6O2

O2

CO2

Respiration: all the time

Photosynthesis: in the presence of light

CarbsCarbs

atmosphere

water

DO < 100% saturated

GPP < R(i.e., -NEP)

atmosphere

water

DO > 100% saturated

GPP < R(i.e., -NEP)

Modeling metabolism: the simple approach

• A free-water approach• Mass balance equation• Many simplifying assumptions• Minimal data requirements

dO2/dt = GPP – R + F + A

Odum, H. T. 1956. Primary production in flowing waters. Limnol. Oceanogr. 1: 103-117.

Gross primary production

Gross primary production

Ecosystem respirationEcosystem respiration

Atmospheric exchangeAtmospheric exchange

All other fluxes, e.g., loads, exports, transfer between thermal strata

All other fluxes, e.g., loads, exports, transfer between thermal strata

Observed oxygen data from sensors

Observed oxygen data from sensors

dO2/dt = GPP – R + F + A (Odum 1956)

R = – dO2/dt + F + GPP + A

GPP = dO2/dt + R – F + A

NEP = GPP– R

NighttimeNighttime

DaytimeDaytime

From night time From night time

Odum, H. T. 1956. Primary production in flowing waters. Limnol. Oceanogr. 1: 103-117.

Crystal Bog Lake 2008

6

6.5

7

7.5

8

8.5

220.0 220.5 221.0 221.5 222.0 222.5 223.0 223.5 224.0

Day of year

DO

(m

g/L

)

DO

DOsat

R = – dO2/dt + F + GPP + A

R = – dO2/dt + F + GPP + A


6

6.5

7

7.5

8

8.5

220.5 220.7 220.9 221.1 221.3 221.5

Day of year

DO

(mg/

L) DO

DOsat

Why add F?

Imagine placing a barrier over the lake to prevent atmospheric exchange. The change in oxygen, driven exclusively by R, would look more like the red line.


6

6.5

7

7.5

8

8.5

220.0 220.5 221.0 221.5 222.0 222.5 223.0 223.5 224.0

Day of year

DO

(m

g/L

)

DO

DOsat




6.5

6.7

6.9

7.1

7.3

7.5

7.7

7.9

220.0 220.2 220.4 220.6 220.8 221.0

Day of year

DO

(mg/

L) DO

DOsat

Why subtract F?

Imagine the barrier again… F artificially increases GPP by driving DO toward saturation

Why subtract F?

Imagine the barrier again… F artificially increases GPP by driving DO toward saturation

Why add R?

If R were somehow turned off, the increase in DO would have been greater.

Why add R?

If R were somehow turned off, the increase in DO would have been greater.

atmosphere

water

F(mg/L/d) = k(m/d) * ( DOsat(mg/L) – DOobs(mg/L)) / z (m)

epilimnion

z = mixed layer depth (e.g., 2 m)

k = piston velocity, or the depth equilibrated per day (e.g., 0.5 m/d)

k = f(wind speed, water temperature)

atmosphere

water

epilimnion

3. Mixed layer depth (atmospheric exchange)

3. Mixed layer depth (atmospheric exchange)

Data requirements for the simple model(sampled at least hourly)

1. Dissolved oxygen1. Dissolved oxygen

2. Water temperature (gas solubility)

2. Water temperature (gas solubility)

5. Barometric pressure or altitude (gas solubility)

5. Barometric pressure or altitude (gas solubility)

4. Wind speed or 0.45 (atmospheric exchange)

4. Wind speed or 0.45 (atmospheric exchange)

-160

-120

-80

-40

0

40

80

0 40 80 120

0

40

80

120

160

0 40 80 120

-160

-120

-80

-40

0

40

80

0 5 10 15 20 25

0

40

80

120

160

0 5 10 15 20 25

mm

olO

2m

-3d-1

R

NEP NEP

GPP

A

B

C

D

DOC (mg L-1) TP (g L-1)

r = 0.70

r = – 0.48

r = 0.68

r = 0.33

Hanson, P.C., Bade, D. L., Carpenter, S. R., and T. K. Kratz. 2003. Lake metabolism: Relationships with dissolved organic carbon and phosphorus. Limnol. Oceanogr. 48: 1112-1119.

Examples of surface water metabolism rates from 25 lakes in northern Wisconsin

80 1 mg L-1 d-180 1 mg L-1 d-1

Metabolism Recipe (simple)

1. RTS for each time step (TS) at night…1. Calculate FTS using difference equation and Cole and

Coraco (1998)2. RTS is the (-) change in DO plus (+) the F

2. GPPTS for each time step during day light…1. Calculate FTS using difference equation and Cole and

Coraco (1998)2. GPPTS is the (+) change in DO plus (+) R24 minus (-) FTS

3. R24 = some integration of RTS over 24 hours

4. GPP24 = some integration of GPPTS during daylight hours

5. F24 = some integration of FTS over 24 hours

Cole, J. J., and N. F. Caraco. 1998. Atmospheric exchange of carbon dioxide in a low-wind oligotrophic lake measured by the addition of SF6. Limnol. Oceanogr. 43: 647-656.

Issues and Assumptions1. Model

1. Atmospheric exchange (F) model2. Buoy measurements representative of the

ecosystem3. Biological model underlying GPP and R4. A, or everything that’s not F, GPP, or R

2. Calculation of the simple model1. Integration from GPPTS and RTS to GPP24 and R24

2. Availability of wind speed, barometric pressure, mixed layer depth

3. Daytime R = Nighttime R

(From: Cole and Caraco 1998)

Issue

Below wind speeds of about 3 ms-1, there is high uncertainty in the estimate of k. This is a problem for most small lakes.

Issue

Below wind speeds of about 3 ms-1, there is high uncertainty in the estimate of k. This is a problem for most small lakes.

Cole, J. J., and N. F. Caraco. 1998. Atmospheric exchange of carbon dioxide in a low-wind oligotrophic lake measured by the addition of SF6. Limnol. Oceanogr. 43: 647-656.

Dissolved Oxygen (mgL

8

9

10

11

12

13

140

7

8

9

10

11

12

Day Night NightDay

(a)

(b)

Elapsed time (hours)

0

12 24 3618 32 426

Dis

solv

ed o

xyge

n (m

g L-1

)

High macrophyte density

Low macrophyte density

littoral

pelagic

Lauster, G. H., P. C. Hanson, and T.K. Kratz. 2006. Gross primary production and respiration differences among littoral and pelagic habitats in North Temperate lakes. Canadian Journal of Fisheries and Aquatic Sciences. 63(5): 1130-1141.

Issue

Different habitats within the lake have different metabolic rates, and sometimes this can be very important.

Issue

Different habitats within the lake have different metabolic rates, and sometimes this can be very important.

anaerobic respiration

PPR

photo-oxidation

Internal waves

strata/sediment exchange

littoral-pelagic exchange

ground water loads

surface water loads Atm exchange

deposition temperatureirradiance

factors affecting DO measurements

Figure 2

dO2/dt = GPP – R + F + A+ E

Issue

If we eliminate A and E from the equation and we assume we know F, then all real processes in A and E are subsumed by GPP and R.

f

0 +

Stylized frequency distribution of R estimates from one week of DO data

Negative R not biologically possible

Some modes clearly represent non-biological processes

How complicated does the biological model need to be?

Examples of added complexity:

1. GPP could be a linear or non-linear function of irradiance

2. R daytime could be a function of irradiance

3. Photo history of algae could affect their GPP or R

Dis

solv

ed o

xyge

n (m

g L-

1)

Day 1 Day 2

Table 2. Equations for the model and nomenclature and definitions for the observed data and free 1

parameters. The governing equation is number 1, and the remaining equations define processes. 2

1. 1111 ttttt FRGPPDODO 3

2a. IPIGPP 4

2b. )/(max

max1 PIIPePGPP 5

3a. 0RR 6

3b. IRIRR 0 7

3c. IRIRR h 0 ;

2

01,

ti

i

iIbi

Ibtth eIeII 8

4. zDODOkF satO /2 9

Observation data and calculated data 10

I = photosynthetically active radiation (mmol m-2 s-1) 11

z = depth of the mixed layer (m) 12

DO = observed DO (mg L-1) 13

DOsat = DO saturation calculated as a function of water temperature (mg L-1) 14

kO2 = piston velocity of DO (m d-1), after Cole and Caraco (1998) 15

Free parameters 16

Param. Description Units Range IP primary productivity per unit of PAR mg O2 L

-1d-1 * (mmol I m-2 s-1)-1

0.0 – 1.0

Pmax maximum primary productivity mg O2 L-1d-1 0.1 – 50.0

R0 night time respiration mg O2 L-1 d-1 0.0 – 10.0

IR respiration per unit of PAR mg O2 L-1d-1 *

(mmol I m-2 s-1)-1 0.0 – 0.1

Ib light decay coefficient 4.0 – 10.0 17

Hanson, P.C., S.R. Carpenter, N. Kimura, C. Wu, S.P. Cornelius, and T.K. Kratz. 2008. Evaluation of metabolism models for free-water dissolved oxygen methods in lakes. Limnol. Oceanagr. Methods. 6:454-465.

Irradiance

Gro

ss P

rim

ary

Prod

ucti

vity

, Res

pira

tion

0

0

R0

P0

IP

Pmax

IR

Simple modelComplicated model(s)

Figure X. Responses for ecosystem GPP and R as a function of irradiance. Parameters are per Table X. (From Hanson et al. 2008)

Hanson, P.C., S.R. Carpenter, N. Kimura, C. Wu, S.P. Cornelius, and T.K. Kratz. 2008. Evaluation of metabolism models for free-water dissolved oxygen methods in lakes. Limnol. Oceanagr. Methods. 6:454-465.

I originalBeta = 4Beta = 6Beta = 8

Day of year

PAR

(µ

mol

m-2 s

-1)

Hanson et al. 2008

Table 2. Processes and free parameters included in each model. Model complexity, in terms of 1

number of free parameters estimated, generally increases with model number. An “X” indicates 2

that the process and any related parameters are included in the model. Equations refer to Table 1. 3

Processes: GPP R F

Parameters: IP Pmax R0 IR Ib

Model 1 Eqs. 2a, 3a, 4

X X X

Model 2 Eqs. 2b, 3a, 4

X X X X

Model 3 Eqs. 2b, 3b, 4

X X X X X

Model 4 Eqs. 2a, 3c, 4

X X X X X

Model 5 Eqs. 2b, 3c, 4

X X X X X X

4

Table 3

From the Word document Hanson et al. 2008

7.0

8.0

9.0

234 235 236 237 2388.5

8.6

8.7

8.8

GraphResults.m

Dis

solv

ed o

xyge

n (m

g L

-1)

Day of year

A) Crystal Bog

B) Trout Lake ObservationM 1M 2M 3M 4M 5

Hanson et al. 2008

With metabolism model prediction, much variance remains unexplained

x 10-3

x 10-4

Dep

th (

m)

Dep

th (

m)

234 235 236 237

A) Crystal Bog Lake

B) Trout Lake

Stab

ilit

y (m

-1)

Stab

ilit

y (m

-1)

6

7

8

9

2221

232425

DO

(m

g L

-1)

T (°C

)

T

DOCrystal Bog Lake

Day of yearHanson et al. 2008

What are the other controls over DO at short (minutes-days)

time scales?

Sparkling Lake (2004)

Bu

oya

ncy

fr

equ

ency

ZI

Win

dT

emp

.D

O

Day of year(total of 50 days)

Tim

e sc

ale

Tim

e sc

ale

Time Time

Signal per scale Details per scaleSparkling Lake Dissolved Oxygen

~1 hour

~1 day

Sparkling Lake, 2004

July 2004 August

DO Buoyancy Frequency

Time Time

Wavelet transforms

Wavelet transforms Wavelet

transforms

Wavelet transforms

Neural networks

Neural networks

For 20 lakes, DO correspondence with…

Irradiance

Langman, O.C., P.C. Hanson, S.R. Carpenter, K. Chiu, and Y.H. Hu. In review. Control of dissolved oxygen in northern temperate lakes over scales ranging from minutes to days. Aquatic Biology.


Temperature

Langman et al. in review


Wind Speed

Langman et al. in review

So environmental data are noisy!

How often and for how long do you need to measure DO to be confident in the metabolism

estimate?

0 1 2 3 40

5

10

188 190 192 1947

8

9

10

182 184 186 1887.5

8

8.5

9

174 176 178 1800

5

10

0 1 2 3 40

1

2

3

4

5

0 1 2 3 4-0.5

0

0.5

0 1 2 3 4-5

0

5

10

0 1 2 3 40

5

10

0 1 2 3 40

5

10

DO

(m

g L

-1)

Req

uire

ddu

rati

on (

days

)

Day of year

Sampling period (hours)

A B C

D E F

GPPRNEPFatm

Figure X. Dissolved oxygen time series (A-C, note differing y axis scales), metabolism (D-F, note differing y axis scales), and required sample duration (G-I) in three study lakes (columns). Metabolism values are means calculated at different sampling periods. Required sample duration is the number of days required to sample at the specified sampling period to detect metabolism within 20% of the mean with a power of 80%.

Day of year Day of year

GP

P, R

, NE

P (

mg

L-1d-1

)

Sampling period (hours)Sampling period (hours)

Little Arbor Vitae Lake Sparkling Lake Trout Bog Lake

G H I

observedsaturation

Staehr et al. In process

Summary

• Environmental data are noisy• A simple metabolism model can work• Data requirements are minimal• Metabolism field is changing rapidly

Documents

Lake metabolism modeling from sensor network data