With the arrival of the IMF in late 2010, the then Irish government published a Four Year Plan for National Economic Recovery. A key assumption is Nominal GDP growth rates for the years 2011-2014. (“Nominal” means not adjusted for inflation.) The plan contains pessimistic and optimistic growth scenarios. Are these numbers realistic?

The history of Irish 4-year Nominal GDP  growth is shown below (uncompounded  e.g. the rate for 1961 is the sum of  rates for 1961,1962,1963 & 1964). The two scenarios from the Four Year Plan are grafted on the end.

How likely is it that the outcome will lie within the expected forecast range? One way to answer this question is to treat historical growth rates as random numbers drawn from a probability distribution. The empirical 4-year growth probability distribution (along with the goverment’s forecast range) is shown below.

The shaded area gives the probability that the 2011-2014 outcome is between the governments pessimistic and optimistic projections. The probability is about 40%. A less than 50% chance that growth falls between the two limiting cases is poor. The Four Year Plan is not realistic in the sense that the range of uncertainty in the forecasts should be increased.

There is some good news here. A more realistic plan will include more optimistic growth scenarios.

vs

Bees are efficient pollinators, and play a critical role in agriculture and natural ecosystems. Recently, researchers at Queen Mary College London reported experiments suggesting that bees are even more efficient than previously thought. When faced with a new arrangement of flowers, a bumblebee is able to determine the shortest path linking the flowers. In effect, tiny bee-brains solve a complex Traveling Salesman Problem (TSP) in a matter of seconds on the fly. This is surpising because TSP is a notorious “hard” problem in optimization. A TSP with N nodes means finding the shortest out of O(N!) possible paths. For 30 flowers, thats about 10³² paths, most of which are very inefficient and a waste of bee energy.

Suppose the bumblebee arrives in the North-East corner of a flower bed which consists of 300 randomly distributed pink, orange and yellow flowers.

The shortest tour solution on the right was obtained using the fast and exact Concorde TSP solver. This took 5 seconds on a MacBook Pro 2.4GHz Intel Core 2 Duo. The optimal tour length ≈13 which is less than 10% the length of a typical randomly chosen tour.

The processor in this MacBook Pro weighs about 20g. A bumblebee weighs about 200mg, with a brain not more than 1% of that, say 2mg. If the bumblebee is really finding the optimal tour, then gramme for gramme, the humble bumblebee brain appears to outperform the MacBook by a factor of order 20,000!

To save the MacBook’s blushes, let’s hope the bee is doing something simpler. For example, using the nearest neighbour algorithm (visit the nearest flower not already visited). In the above example, the nearest-neighbour tour length is ≈16, which gets the bee most of the way there in terms of efficiency. This algorithm, while not exact, takes only 0.02 seconds on the MacBook Pro.

By the way, Charles Darwin was aware of the surprising power of small insect brains. “It is certain that there may be extraordinary activity with an extremely small absolute mass of nervous matter.. the brain of an ant is one of the most marvellous atoms of matter in the world, perhaps more so than the brain of man.“ – Charles Darwin, Origin of Species, 1859

Here is the R code used above.

library("TSP")
concorde_path("/usr/local/bin")
Nflowers <- 300
garden <- matrix(runif(Nflowers*2),Nflowers,2)
garden[1, ]<-c(1,1)
garden.dist <- dist(garden)
garden.tsp <- TSP(garden.dist)
garden.path <- solve_TSP(garden.tsp,method = "concorde")
garden.tour <- as.vector(TOUR(garden.path))
ex=par(mfrow=c(1,3))
plot(garden,xlab="flower garden",ylab="",pch=16,col=c("orange","pink","yellow"),xaxt="n",yaxt="n",cex=1.5)
points(garden, pch=1,cex=1.5)
plot(garden,xlab="concorde",ylab="",pch=16,col=c("orange","pink","yellow"),xaxt="n",yaxt="n",cex=1.5)
lines(garden[garden.tour,],col="black",lwd=2)
points(garden, pch=1,cex=1.5)
par(ex)


A handy script to help  with the external installation of concorde TSP on the Mac is available here. The TSP package also needs to be installed.

]]> http://joewheatley.net/bee-vs-macbook-pro/feed/ 0 http://joewheatley.net/survey/ http://joewheatley.net/survey/#comments Tue, 20 Jul 2010 16:37:22 +0000 joe http://joewheatley.net/?p=2315 If you are a professional with interest in weather/climate impacts in agriculture Biospherica would like to hear your views.

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The output from the U.S. National Centers for Environmental Prediction (NCEP) Global  Forecast System (GFS) is freely available. The surface resolution of the model is ≈ 0.3º× 0.3°. The model runs every 6 hours, producing forecasts at 3-hourly intervals extending out to 16 days. As an example of output from GFS, the map (below) shows the predicted average  temperature at 2 metres over the entire globe for the next 24 hr (date of this post). The map shows predicted cold conditions in Europe, and the continuing heatwave in Australia.

How the map was made

GFS forecasts are in a format called GRIB2. According to Wikipedia, “GRIB (GRIdded Binary) is a mathematically concise data format commonly used in meteorology to store historical and forecast weather data.” GRIB files contain physical fields such as temperature, humidity etc defined on a spatial grid, as well as boundary conditions such as vegetation type and elevation. The data might be assimilated from observations, or output from a forecast model.

The first step is to translate the GRIB into a raster format such as netcdf which can be read in R. For example, the GRIB2 file gfs.2009121700/gfs.t00z.sfluxgrbf03.grib2 contains the 3-hr forecast surface data on 17 Dec 2009  produced at 00 UTC (midnight universal time). An inventory of the data contained in this file can be seen here. Download this forecast as temp.grb

loc=file.path("ftp://ftp.ncep.noaa.gov/pub/data/nccf/com/gfs/prod/gfs.2009121700/gfs.t00z.sfluxgrbf03.grib2")
download.file(loc,"temp.grb",mode="wb")

To read temp.grb a utility called wgrib2 needs to be installed on your system. Then data such as land fraction can extracted into a netcdf file LAND.nc using the R shell command

shell("wgrib2 -s temp03.grb | grep :LAND: | wgrib2 -i temp00.grb -netcdf LAND.nc",intern=T)

The ncdf package can now be used to read the contents of LAND.nc.

library(ncdf)
landFrac <-open.ncdf("LAND.nc")
land <- get.var.ncdf(landFrac,"LAND_surface")
x <- get.var.ncdf(landFrac,"longitude")
y <- get.var.ncdf(landFrac,"latitude")

The 1152×576 matrix land takes values 1 for land and 0 for water (sea-ice is 1). x and y are the longitude and latitude of the non-uniform GFS grid.

2m temperature data can be read in the same way. The average of the first 8  forecasts was called t2m.mean and plotted using image.plot() from the fields package:

library(fields)
rgb.palette <- colorRampPalette(c("snow1","snow2","snow3","seagreen","orange","firebrick"), space = "rgb")#colors
image.plot(x,y,t2m.mean,col=rgb.palette(200),axes=F,main=as.expression(paste("GFS 24hr Average 2M Temperature",day,"00 UTC",sep="")),axes=F,legend.lab="o C")
contour(x,y,land,add=TRUE,lwd=1,levels=0.99,drawlabels=FALSE,col="grey30") #add land outline

warming in the satellite era

Data from satellites have been used to construct a global temperature record beginning in December 1978. This covers the recent warming period. The University of Alabama at Huntsville (UAH) 2LT product gives average temperature of the lower atmosphere derived from microwave radiances. Planck’s law says that microwave radiances are related to temperature.

The plot below shows UAH monthly time-series and a least-squares regression fit. The slope of the regression line is 0.013ºC/year (equivalent to 0.13°C/decade or 1.3°C/century). Also shown is the Gistemp global temperature time-series over the same time interval. Gistemp is based on surface weather stations. The trend in the Gistemp series is higher, 0.18°C/decade.

uncertainty in trend

A linear regression fit to global temperatures is a statistical model in which strong residual variations are superposed on a linear warming trend. The residual variations reflect “noise” in the climate system. ENSO, random events such as volcanos, and many other modes of variability contribute to the “noise”. The “uncertainty” in the global warming trend can be estimated in the context of this statistical model.

From this point of view, the UAH monthly data are one realisation of a random sample of length N. In reality we only have one sample, the UAH data from the real world. However, if repeated sampling and slope measurement were possible, it would yield a range of outcomes due to noise. The width of the resulting slope probability distribution reflects the uncertainty in the global warming trend. It is evident that there is strong serial correlation in the residuals. This enhances the uncertainty, because it reduces the effective number of independent degrees of freedom in the data. Fewer degrees of freedom means more uncertainty.

The UAH residuals are fit to an auto-regressive model to account for serial auto-correlation. The Akaike information criterion to choose an optimal order n. For UAH it  turns out that n=2, with coefficients 0.58  and 0.24. To find the slope probability distribution 10,000 time-series were used in a Monte Carlo simulation. Each simulated time-series assumes a UAH trend-line of 0.13°C/decade and an AR(2) noise term parameterized using the AR(2) model fit to the UAH residuals. Linear regression fit on each of these simulated time-series yields the slope probablity distribution.

A typical result is shown above along with a fit to the normal distribution. By construction the mean is 0.13°C/decade as expected. The standard error is 0.036°C/decade. This is 3.4 times greater than the standard error with independent and identically distributed (iid) residuals.

The 2σ confidence interval for UAH global temperature trend is 0.05 to 0.2°C/decade. Unfortunately, this is quite a wide window. Would another 30 years of data help? Assuming that the properties of the UAH time-series do not change, and extending the above simulation for another 30 years, the standard error is reduced from 0.035°C/decade to 0.026°C/decade. Better, but still a surprising degree of uncertainty.

R code

The above analysis was carried out using R functions ar() and arima.sim()

fitLin <-lm(GLOBAL~dates) # linear regression fit to GLOBAL= UAH global temperature, dates = UAH dates
ar.fit <- ar(residuals(fitLin)) # AR model for residuals
N=10000; # number of simulated UAH time-series
fits <- cbind(rep(0,N),rep(0,N))
trend <- coefficients(fitLin)[1] + dates*coefficients(fitLin)[2] # UAH trend line
#Monte Carlo
for( i in 1:N){
warming <- trend+arima.sim(n = NData, list(ar = c(ar.fit$ar[1], ar.fit$ar[2])), sd=sqrt(ar.fit$var.pred)) #generate sample
fit.warm <- lm(warming~dates) # linear fit
fits[i,1] <- coefficients(fit.warm)[1];
fits[i,2] <- coefficients(fit.warm)[2]; # slope of simulated data
}
library(MASS) # MASS library must be installed
gf <- fitdistr(fits[,2],"normal") # fit a normal distribution to the slopes
mn <-gf$estimate[1] #mean
sig <- gf$estimate[2] #standard error


Respiration CO2 derives from maintainance and growth respiration by vascular plants (“autotrophic” respiration) and also by the decay of organic matter in soil and litter layers (“heterotrophic” respiration). At the ecosystem level, the net exchange of CO2 with the atmosphere is called Net Ecosystem Exchange (NEE). NEE is just the difference between total ecosystem respiration (RE) and photosynthesis (or Gross Ecosystem Production, GEP) :

At night, GEP and the flux of CO2 from the ecosystem to the atmosphere equals RE. During daylight hours, GEP switches on and NEE is normally negative during the growing season. Of course NEE depends on sunlight, air temperature, soil moisture etc. Fortunately this important property of ecosystems is directly measurable.

Eddy Covariance

Under normal conditions, air motion above vegetation is turbulent. This fact is the basis of a statistical technique called eddy covariance which measures the flux of CO2 between ecosystem and atmosphere. A setup similar to the one shown on the left is mounted on a tower rising above the top of the vegetation canopy. The setup consists of a gas analyser (measuring instantaneous CO2 concentration), and an anemometer (capable of measuring the instantaneous vertical component of the wind velocity).

To a good approximation, the CO2 flux is just the covariance of the vertical wind speed with the CO2 concentration . For example, if and are uncorrelated, the flux is zero. The covariance can be obtained by recording data at high frequency over 30min intervals, say. This gives a time-series of CO2 fluxes at half-hour intervals. The eddy covariance technique gives information on NEE on a spatial scale which is typically . Of course, the technique also works for other trace gases or water vapour. There is a good wikipedia article on eddy covariance which provides additional details.

According to Fluxnet, there about 500 flux towers making continuous eddy covariance measurements of NEE worldwide. Given the diversity of Earth’s ecosystems, this is a small number. Flux tower data is rare and valuable.

NEE Data

With this technical explanation out of the way, we get to look at NEE for some real ecosystems. Access to (mainly North American) flux tower data was obtained through the ameriflux network. Two different forest ecosystems are compared. Harvard forest is a 1200Ha temperate broadleaf deciduous forest in Massachussetts. This secondary growth forest has been studied intensively since it was established in 1907.^[1] A 30m flux tower has measured NEE at Harvard Forest since 1993. km 67 Sanatarem flux tower on the other hand is located in primary tropical rainforest, Tapajos National Forest, Para State, Brazil. Three years of data are available 2000-2003 from this 64m tower.

Half-hourly time-series of CO2 fluxes were generated as shown below using the statistical programming language R.

The qualitative features of CO2 fluxes are as expected. Namely, both forests lose carbon at night while daytime fluxes tend to be negative, temperate forest shows a strong seasonal signal and there is a much weaker wet/dry seasonal signal in the tropical forest. There are some surprises, however. Peak summer carbon fluxes at Harvard Forest are as large as Santarem tropical forest fluxes . Another surprise is that there is stronger carbon absorbtion by the tropical forest during the dry season, which seems to contradict intuition about dry season water stress.^[2] Perhaps the biggest surprise is the relative performance of the two forests as net carbon sources or sinks.

Sources or Sinks?

Cumulative NEE shows whether these forests are sources or sinks of carbon. This is simply obtained by applying R‘s cumsum() function to the half-hourly time-series above. The graph (below) shows that Harvard forest has been a strong and even accelerating sink for CO2 (= 2tC/Ha/y) since 1992, even though it is 100 years old. By contrast, primary forest at the Santarem site was a source of CO2 between 2002 and 2005. Researcher suggest this may be due to the presence of excess dead wood in the area following earlier disturbances e.g. 1998 El Nino drought.^[2]

It is remarkable that Harvard Forest is still an agressive carbon sink (0.25KgC/m2/y) after 100 years of growth.

Conclusions

The above illustrates some of the surprises and complexity of real ecosystem data. When memory effects are large, intuition can be a very poor guide. Models, such as those used in long-term climate research, are necessarily simplifications of reality.

This is the R code used to download, process and plot the flux tower data. In a follow-on post an ecosystem model for Havard Forest NEE will be built in R.

References

[1] Forests in Time: The Enviromental Consequences of change in New England, by D. Foster and J. Aber, 2004

[2] Carbon in Amazon Forests: Unexpected Seasonal Fluxes and Disturbance-Induced Losses, Saleska et al, Science vol. 302 2003

]]> http://joewheatley.net/flux-towers-part-i/feed/ 1 http://joewheatley.net/piling-on-the-carbs/ http://joewheatley.net/piling-on-the-carbs/#comments Wed, 08 Jul 2009 19:26:22 +0000 joe http://joewheatley.net/?p=569 Human activty is putting nearly ten billion tons of additional CO2 into the atmosphere each year. These emissions derive from burning of fossil fuels and changes in land use. Only half of this additional CO2 stays in the atmosphere, however. The rest is absorbed by oceans and by land (vegetation + soils). Climate researchers investigate how these complex carbon reservoirs (atmosphere, ocean and land) operate under future climate change scenarios.

So far vegetation have responded positively to increased CO2, a fact that is sometimes called “carbon fertilisation”. The data suggest that terrestrial ecosystems have absorbed more than 90 Gt (Gt = 1 billion metric tons) of additional CO2 since 1959. 90 GtC equivalent is a lot of vegetation. For comparison, all the world’s tropical rainforests = 200GtC equivalent. Even without any greenhouse effect, anthropogenic CO2 emissions have already had a large impact on terrestrial ecosystems.

CO2 record

Atmospheric CO2 has been routinely recorded at Mauna Loa since 1958 and is now recorded at many other locations worldwide. Monthly average data from Mauna Loa are available from the Scripps Institute. The Carbon Dioxide Information and Analysis Center (CDIAC), Oak Ridge National Laboratory, provide historical fossil fuel emissions data, and emissions associated with landuse change. CDIAC also have atmospheric CO2 data obtained from ice core studies, for example from Law Ice Dome in the Antarctic.

The “Global Carbon Budget” from 1959 is summarized by Le Quéré at the University of East Anglia. The fraction of CO2 emissions which remain in the atmosphere (called the Airborne Fraction) is shown in the top chart of the figure below. The mean value of the Airborne Fraction is 43%. The linear fit suggests that a slightly greater proportion of CO2 is remaining in the atmosphere now than in the past.

Le Quéré also gives results of a global ocean calculation of annual CO2 absorbed by the oceans, based on observations.  In this estimate, oceans absorb about 29%  of emissions on average. Whatever is left over (27%) must equal the amount absorbed by terrestrial sinks. In the ocean model, there has been a decrease in the ability of the cold Southern oceans to absorb CO2 (centre chart). This means there is a downward trend in the ocean uptake fraction. This explains the upward trend in the Airborne Fraction. The remaining terrestrial fraction (bottom chart) is highly variable, but has no significant trend.

Two more things are worth noting from the above figure.

(1) the terrestrial uptake is similar in magnitude to the ocean uptake even though only 25% of the surface area of the earth is covered by vegetation.

(2) the terrestrial uptake is much more volatile than the ocean uptake. In some years, vegetation was a source of CO2.

Why is the variability of the terrestrial carbon fluxes so large? The answer is that this reflects vulnerability of vegetation to climatic variability, particularly droughts and fires. The negative effects of the 1987/8 and 1998 El Nino events are obvious on the bottom chart.  The 1991 eruption of Mount Pinatubo apparently lead to increased CO2 uptake.[2]

Accumulation of Carbon by Vegetation

The net cumulative uptake of CO2 by vegetation since 1959 is shown on the left hand plot below. Again this is derived from one specific model of the ocean uptake.

The right-hand plot shows the relation between terrestrial accumulation and cumulative emissions since 1959. This is accurately linear. In some climate models, future global warming causes terrestrial ecosystems to degrade and eventually become net sources of CO2. If this starts to occur, the right hand plot would begin to flatten out.

Where is the extra Vegetation?

For a long time it was believed that the primary terrestrial carbon sink was in growing Northern forests. However more recent work suggests that about 1 Gt of additional CO2 is absorbed per year by mature Tropical Rainforest, as well as 1GtC per year in Northern forests.[3] Of course, deforestation of tropical forests is also the major source of  “land use change” emissions.

It is remarkable that so much uncertainty surrounds such a basic issue. The arrival of new CO2 sensing satellites may improve this situation in the near future.

References

[1]Saturation of the Southern Ocean CO₂ Sink Due to Recent Climate Change, Le Quéré et al http://www.sciencemag.org/cgi/content/abstract/1136188

[2] Anthropogenic and biophysical contributions to increasing atmospheric CO2 growth rate and airborne fraction, M. R. Raupach et al http://www.biogeosciences-discuss.net/5/2867/2008/bgd-5-2867-2008.pdf

[3] Missing carbon mystery: Case solved? Nature Report Climate Change Jane Burgermeister

Equinox Stone, Hag's Hill, Ireland

Biospherica is a commentary on the condition of the terrestrial biosphere, in particular vegetation. More accurately, it discusses analysis and applications of what are often referred to as “bio-geophysical data”. Unfortunately there are many barriers to exploiting  bio-geophysical data effectively. Data are spread over a number of different platforms, and analytical techniques can be quite specialised. Biospherica is about breaking down these barriers. In future these data will play a much greater role in agriculture, trading, conservation policy, carbon markets etc than it does today.

An important class of bio-geophysical data is based on reflectance of sunlight from the Earth’s surface. Solar reflectance is measured by instruments carried on board Earth observation satellites. The spectrum  (i.e. how the reflectance depends on wavelength) at a particular surface location gives information about density of live vegetation at that location. Since the 1990s there has been an explosion in the collection and archiving of such data. In fact the quantity of data collected per unit time on the Earth’s biosphere greatly exceeds the quantity collected in financial markets, where prices  of 100,000′s of financial securities are tracked and stored in real-time.

Biophysical data are not as organised or accessible as financial data, and their application less transparent. Financial data include everthing from stock trade data, to commodity option prices  or over-the-counter derivatives quotes. To extract meaningful information about the overall state of financial markets from such a large and complex dataset, stock indices such as Standard & Poors appeared long ago. Other indices have become important more recently, such as the VIX volatility index. Of course many reports and tools covering every imaginable aspect of financial markets are readily available.  Analysis of bio-geophysical data, on the other hand, is typically carried out by large public agencies, such as the US Department of Agriculture or the Joint Research Centre of the EU Commission.

One important theme of Biospherica will be to explore interesting parallels between the terrestrial biosphere and financial markets. Both are complex, non-linear systems with feedbacks, thresholds and instabilities. It is no accident that the first organised derivatives markets were in agricultural commodities, which helped participants to reduce their risk. Natural climatic variability is a major influence on vegetation and agriculture, whereas economic cycles influence financial markets. Secondly, an estimated 90% of the vegetative cover of the earth has been modified by human activity to a significant extent ^[1] . It could be said that the ecosystems that make up the biosphere are effectively being managed by human beings, just as a financial portfolio is managed by a fund manager.

The first (real) post in Biospherica will continue the financial market analogy with a post entitled Global Agriculture’s Bull Run.

Hag’s Hill

I want to end this post on an auspicious note, because it is the first. The image (above) was taken at 07:15 GMT 20 March 2009 in Cairn T at the Hag’s Hill, Loughcrew, County Meath, Ireland. It shows the Equinox Stone, so-called because it is illuminated for a brief period as the sun rises due east on the spring and autumn Equinoxes. The hilltop mounds at Loughcrew were built around 3400bc by Neolithic farmers.  At Cairn T, a narrow passageway leads to a small chamber containing the Equinox Stone.  The irradiant glow from the leaf-like solar symbol at Loughcrew is a bewitching sight. We don’t know the precise symbolism that was intended by this light show. However it is easy to believe that it signified the start of the growing season at this northerly latitude (54^o ). 5,000 years ago it must have appeared a very auspicious event, as it still does today.

[1]Anthropogenic Biomes of the World, Ellis and Ramankutty