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Good morning and welcome to this, the lecture number 39 of the course Stochastic Hydrology.
In the last lecture, we essentially discussed about data issues.
Specifically touching upon the data consistency checks and in the previous lecture, we had
talked about the double mass curve for data consistency. And in the last lecture, we also
covered the concept of specific flow. And how we use this concept to examine the consistency
of data, which is in this particular method is especially useful, when we are talking
about large catchments, where a number of gauges are recording the flows, specifically
the flows. And then, we would like to examine whether the flows in fact are consistent with
each other, as recorded from various gauges. And we took up the case study of the Narmada
basin and then, examine at various sub basins how the specific flows compare with each other.
Then, I also discussed in the last lecture, the data representation through box plots.
Box plots essentially will provide a good representation for the uncertainties starting
with including the range, it shows the range between minimum flow and maximum flow, it
also shows the median, it shows the 25 percentile, 75 percentile and sometimes we also put the
mean. Now, these box plots are useful especially,
when we are comparing different time series or the variability of a particular time series
across time across different time window. Let say, that you have a long time series
of 100 years of flows, or you would like to examine how the variability was in the first
20 years as compared to the next 20 years as compared to the next 20 years and so on.
So, these box plots are handy tools for examining such variability in the data.
Then, we also discussed in the previous lecture, how we normalize the flow data, when we have
the gauge data, which are contributed essentially the flows at a particular gauge are essentially
contributed from controlled flows upstream of that upstream of that particular gauge
location; which means, you may have a reservoir or you may have some other control structure
through which the flows are coming and then, you are recording the controlled flows at
this particular location. And for all analysis I have been emphasizing
that, you need naturalized flows or normalized flows, which means you have to take out the
effect of the anthropogenic interventions in terms of the reservoirs or in terms of
at any other let say, lift irrigation schemes and so on. You may have several such anthropogenically
anthropogenic interventions, structural interventions that are taken place in the stream.
And therefore, the recorded data will have signals of the controlled flows, and these
signals have to be removed and then, you convert the flows as natural flows naturalized flows.
And then, deal do your analysis using the techniques that we have covered in in this
course on the naturalized flows. So, we have covered how to convert the flow data at a
particular location into normalized flow data or naturalized flow data that essentially
completes the portion that is set out for this particular course.
So, in today’s lecture what I will do is, this is a special lecture as of last lecture
before we summarize, I will cover a recent application of most of the techniques that
we have covered in this course. And this topic is timely, because in hydrologic research,
most of the most of the research that is going on is in this area. So, I will just give a
broad flavor of how we use the techniques that we have covered in this course, to in
this particular area, and this area is the climate change impacts on hydrology.
So, this is a special lecture, where I will just give a broad flavor I will not go into
too much of details of how we do the modeling and so on. But, given that we have gone through
38 lectures so far, we will know all the methodologies that I will be discussing, so I will not go
into the details of what kind of numbers we get out of methodologies and so on.
So, we will just go through the broad idea of the concepts that are involved and the
modeling aspects that we do using the statistical techniques that we have learned stochastic
techniques that we have learned. So, this is this lecture is on hydrologic impacts of
climate change, and especially on the quantification of uncertainties.
And I have taken these slides from of my PhD students, thesis presentation, his name is
Subimal Ghosh, he is already a faculty member at IIT Bombay after completing his PhD from
IISc Bangalore. So, the slide set follow are essentially taken from his presentation with
permission.
As I mentioned in the previous lecture, the issue of climate change impacts on hydrology
is the one that deals with relating the long term changes that are happening on the climate
with the changes that are likely to happen in hydrology. As I mentioned in the last class,
the global average temperatures are known to be changing known to be increasing. In
fact as a result of which, the sea levels are rising and the precipitation patterns
across the globe are varying are changing. All of these three prominent signals of climate
change have direct implications on hydrology. They will affect the precipitation patterns,
regional precipitation patterns as the result of which, the hydrologic extremes of floods
and droughts are likely to change, the magnitudes are likely to change, magnitudes of the floods,
the frequency of the floods and vulnerability of various regions due to floods, all of these
are likely to be affected. The water availability in a river basin is
likely to be affected, the evapotranspirative demands of the crops are likely to be affected;
and in countries like India, where primarily the water resources systems are operated for
irrigation irrigated agriculture, these will have direct implications. And the rise in
sea levels will have implications on the costal indentation, coastal flooding as well as,
on the salinity ingrates into the ground water. So, the climate change impacts need to be
studied at regional scales, taking into account the projections provided by what are called
as a general circulation models or the global climate models.
So in this lecture, we will see how we do that and also, how we address uncertainties
associated with our projections. Because, standing at this point in time we need to
assess what is likely to happen to let say, the water availability in a particular river
basin over the next 20 years, 30 years, 40y years and perhaps, 1 full century; because,
our water resource systems are designed to operate designed to serve for next about 100
years. And therefore, we would we should know how this systems are likely to perform in
the phase of climate change over the next so many years.
So, what we do is that, we pick up the simulation of climate variables as provided by the general
circulation models. Now, the general circulation models as I mentioned in the previous class,
are essentially models of climate; and these are driven now by the green house gas emissions
and typically the CO 2 levels, CO 2 emissions into the atmosphere and corresponding to the
various levels of CO 2, emissions that are likely to happen in the future.
We have what are called as a scenarios, that is emission scenarios for example, one of
the emission scenario can be double CO 2; that means, in a particular time frame CO
2 levels will be doubled like this we have several scenarios, these are given by I P
C C, the Intergovernmental Panel on Climate Change, I P C C. And for these scenarios,
the climate models provide us the projections of climate variables, several climate variables.
For example, what is likely to happen to temperature, what is likely to happen to relative humidity,
what is likely to happen to pressure levels let say, geo potential height, that is gravity
at the straight pressure levels and so on, and what is likely to happen to sea surface
temperatures, mean sea level pressure and so on. So, there are large number of climate
variables for which the simulation are provided by the global climate models or the general
circulation models. Now, as hydrologist we pick up this simulation
of the climate models and then, relate it with the hydrologic variables. We will be
interested in, what is likely to happen to precipitation in a particular catchment or
what is likely to happen to stream flow at a particular in a particular river basin,
what is likely to happen to evapotranspiration in the particular river basin and so on. These
variables that the hydrologist are interested in, are not well simulated by the climate
models for various reasons we need not worry about those things. But, the variables at
a really of interest to the hydrologist are not well simulated by the general circulation
models. In addition, the general circulation models
operate on large scales by their very nature, because they have to account for the entire
globe, these are circulation patterns. So, which which means, you cannot afford to just
look at one particular region in isolation, everything is related; and therefore, the
global climate models will generate the climate taking into account, various grids across
the globe and because of that, because of the computation requirements, because of the
several other requirements, they essentially talk operate at large scales. What I mean
by that is? The grid size of climate models are much larger compared to the size that
is required for hydrologic impact assessment. So, because of these two major reasons, namely
that the variables that are required in hydrologic applications are not well simulated by the
climate model. And that the global climate models or the general circulation models provide
the outputs essentially at large scales, because of these two reasons we adopt what is called
as the downscaling. So, we downscale from the G C M’s to the
hydrologic scales, how we do this, we will study slightly, we will study in the present
lecture. So, we pick up the simulations of provided by the general circulation models,
we need the hydrologic variables at a local or regional scales; and this we obtained from
downscaling of the G C M output, we use this to project into the future, the hydrologic
scenario. So, associated with the climate scenario, we provide a hydrologic scenario.
And then, look at the risk associated with hydrologic extremes let say, the droughts,
the floods and even the risk associated with operating a particular reservoir or water
resource system in a particular manner. Risk associated with hydropower being lower than
a particular level, water supply being a particular level and so on.
So, the risk associated with not only hydrologic extremes, but also with the various features
of the water resource systems. And we developed risk based approaches and so on. So, this
is a broad idea of what we do in hydrology in as much as climate change impacts are concern.
However, we will focus in the in the lecture essentially on downscaling techniques and
a specific class of downscaling techniques called as a statistical downscaling techniques,
that use the transfer function approaches. When we do all of this analysis, the topics
that you have covered in this particular course and some adaptation of those topics will come
in handy, in addressing the various levels of uncertainties. There are model uncertainties,
because we are relying on the global climate models or the general circulation models,
there are handfuls of such models in fact, there may be as many as 25 such models, for
which the outputs are available across the globe and there are many more coming.
So, we have what is called as a G C M uncertainty? That is, General Circulation Model uncertainty,
depending on which model you choose for the particular region that we are interested in,
you may get different types of projections when you bring it down to the regional levels.
There are various reasons for that, we will not in this lecture we will not worry about,
why the G C M uncertainty in fact, happens. Then, there is also a scenario uncertainty
as I said scenarios are pictures of how the world is likely to evolve in in in future,
now there is a enormous uncertainty associated with the scenarios. How the CO 2 levels will
be in the future and because of which, how the temperatures are likely to be in the future
and so on. Standing at this point in time we are trying to project it into the future
and there are large numbers of uncertainties associated with scenarios.
Then within the model itself, there will be some uncertainties, because of the parameters
and so on. The G C M uncertainty that I talked about, are intermodal uncertainties when you
choose one G C M, it produces a certain type of projection, whereas another G C M produces
another type of projection that is intermodal uncertainty. But, there are also intra-model
uncertainties, which arise because of the parameters that have been put into the G C
M, the way parameters are estimated, the way boundary conditions are estimated, the way
initial conditions are estimated and so on. So, that leads to intra-model uncertainty.
Then as I said, from the climate models we are bringing it down to hydrologic scales
on hydrologic variables and therefore, we are doing a particular type of downscaling.
So, the down scaling itself produces certain uncertainty. So, there are large numbers of
uncertainties when we are talking about climate change impacts on hydrology. The methods,
the tools, the techniques that you have studied in this particular course will all come in
handy, when we are addressing these kinds of uncertainties.
So, in the probabilistic approach I I will I will just give a broad overview of what
we do here and in the particular application that I will be talking about, in this lecture
that will deal with the drought index or how the drought scenario is likely to evolve in
a particular river basin. So, from the G C M we do a statistical downscaling, using the
principle component analysis and linear regression, we also add fuzzy clustering, but I will not
cover that, because we are not covered in the course the fuzzy clustering, but typically
we have studied the regression using the principle components.
So, I will just explain how these techniques are used in that. So, for doing this we need
the G C M output. And the specific example that I will be covering, we will use the mean
sea level pressure, so mean sea level pressure is likely to effect the precipitation directly.
And mean sea level pressure is well simulated by most of the G C M’s and therefore, we
pickup mean sea level pressure as, what is called as a predictor of the precipitation,
which is called as the predictand. We are now developing a relationship between
the predictand, which is a precipitation at a regional scale with the predictor which
is a mean sea level pressure at several grid points, several G C M grid points. From this,
we get the subdivisional precipitation in in this particular case, we are talking about
Orissa subdivisional precipitation as I will show you presently. From this, we get what
is called as the standardized precipitation index of 12 month scale and that is what we
use as drought index. And we address the G C M and scenario uncertainty by fitting probabilistic
probability distributions to the S P I 12. And then, we address several levels of uncertainties,
one is a data sample uncertainty itself, if your sample size is too small size is reasonably
small and you are trying to fit a probability distribution to that; then the probability
distribution itself will have uncertainty, that we address through imprecise probabilities.
And then finally, we provide the probability distributions for the drought index into the
future, so this is a broad overview.
We will go into some of the details here. So as I mentioned, the downscaling deals with
bringing down from a larger scale like this, this is a G C M grid point, it until recently
it used to be of the size 2.5 degree by 2.5 degree, that is 2.5 degree longitude, 2.5
degree latitude, which may be around 250 kilometers by 250 kilometers; that means, we are getting
one output from the G C M for the scale of 250 kilometers by 250 kilometers; whereas
on the hydrologic scale, if you want to run any hydrologic model, grid base hydrologic
model like for example, variable length field trace and capacity model and so on, you will
need this outputs at 15 kilometer by 15 kilometer, 12 kilometer by 12 kilometer and so on.
So, these are the grid sizes required for hydrologic processes. In fact, depending on
the process that we will be interested in, these need to be even smaller of the order
of 5 kilometer by 5 kilometer and so on. So, we must be able to bring down the G C M outputs
to the hydrologic scales, and this we do by what is called as a downscaling.
There are two major ways of doing downscaling, one is called as the dynamic downscaling,
and another is called as a statistical downscaling. The dynamic downscaling is beyond the purview
of the hydrologist. In dynamic downscaling they actually develop the regional climate
models taking into account the the regional climate models, take the boundary condition
from the global climate models; and therefore, for a given G C M, you develop a regional
climate model for a particular region. However, that is beyond the scope of hydrologist,
so we adopt what is called as the statistical downscaling. The statistical downscaling produces
future scenarios based on statistical relationship between the climate variables. Let say, in
this particular case we are talking about mean sea level pressure, so mean sea level
pressure at a particular grid point, we may have mean sea level pressure here, and then
we are interested in precipitation at a smaller scale.
So, we develop statistical relationship between the mean sea level pressures at one or more
grid points, with the precipitation at a particular location. These statistical relationships
can be as simple models as simple linear regression or multiple linear regressions to very very
complex models using the conditional random fields and so on. But essentially, they use
the concepts of probability and statistics. Once we develop this statistical relationships
based on the historical data, we hold this statistical relationship intact and then,
use it for future projections. So, the G C M’s will provide the future
projections, we use the same statistical relationship that is developed based on the historical
data between let say, the M S L P Mean Sea Level Pressure and the precipitation. We hold
this intact and then, use the M S L P as projected by the G C M into the future and then using
the same statistical relationship, we project the precipitation into the future.
The main advantage of statistical downscaling is that, it is computationally simple and
then any new area you can develop a new relationship for different G C M’s, whereas for dynamic
downscaling for every G C M, you have to develop a different regional climate model ok.
We will just look at one application here, this region that is shown is the Orissa meteorological
subdivision as provided by. So, we have a rainfall series available for Orissa meteorological
subdivision, this area shows the Orissa meteorological subdivision and the location is here.
Now, this region is sensitive to climate change, because it is a first of all, it is a coastal
area then, there are increase in hydrologic extremes that are evidenced in the past. It
is subjected to both floods as well as, droughts frequently in the past. Then we use a rainfall
data, available from 1950 to 2003, we have monthly data available. So, this time series
is what is available to us, what we do is that, we superpose the G C M grid points,
this is a particular G C M I will show you which G C M we are using and so on. So, you
choose a G C M and then superpose on this area, the g c m.
If you have historical data at all this grid points on the climate variables that we have
chosen as predictors, then we use the historical data to develop a relationship between the
particular predictors at this various locations with the precipitation at this regional scales.
So, we have the precipitation at this regional scale which is called as a predictand, we
want to predict that particular variable; and we have the climate forcing coming from
what are called as the predictors. So, the predictor in this particular case is, the
mean sea level pressure, so the mean sea level pressure values we may have at various grid
points like this. In the absence of observed predictor value,
data on the predictor variables we use what are called as a reanalysis data. The reanalysis
data essentially provide us the closest data to the historical point, they use a large
number of data sets across the globe and then, they run idealized G C M and then, provide
us with the data on all the climate variables. So, we will have the reanalysis data and we
also have the G C M produced data for the future.
We use the reanalysis data or the historical data to develop relationship between the predictand,
which is the rainfall here and the predictors at various points and then, hold this statistical
relationships intact; and then, look at the projections provided by the G C M, use the
same statistical relationship and provide the projections of the precipitation, that
is a principle of statistical downscaling.
So, in this particular case, we use the G C M of C C S R, Japan, this is a A G C M;
and then, we provide we used one of the scenarios just for demonstration I will use this scenario
now, it is called as B 2 scenario, it is from the third assessment repowered of I P C C,
Intergovernmental Panel on Climate Change. We use the climate predictor as M S 1 P to
begin with Mean Sea Level Pressure and we relate the mean sea level pressure with the
precipitation at the Orissa metrological subdivision. And the G C M output are provided up to 2100,
so you have certain output for calibration of your models as well as, you have the projections
into the future, so we use from 1950 to 2100. You look at this now, at each of these grid
points you have one variable mean sea level pressure available at this locations and then;
that means, sea level pressure is a time series, its available in the form of time series.
So, we use the monthly data of rainfall from 1950 to 2003. So, you have the time series
of the rainfall, you have the time series of the mean sea level pressure at all of these
locations, you can develop a relationship between the rainfall and this predictors.
Because of the size of the problem as you can see here, if you take 4 by 4 it will be
16. And then as your predictor number of predictor’s increases, you will have larger number of
a larger number of variables involved in the regression, if you are using regression. Because
of the size as well as, because of what we call as multicollinearity which I discussed
in the multiple linear regression topic, we do the principle component analysis to remove
the multicollinearity and to reduce the size. So, we use the principle component analysis
to identify how much of percentage of variance, that is explained by the principle components.
So, in this particular case, you can see first two explained almost about 95 percent, this
is let say 59 and this is 37, so about 96 percentage will be explained, but we can take
up to third, upto the third principle component you choose for modeling purposes; all other
principle components, do not add to the explanation of variance significantly.
So, we choose three principle components. Recall your topics that we covered in the
multiple linear regression using the principle components, we use the eigen eigen values
and eigenvectors; and then, identify those particular principle components initial few
principle components, which explain most of the variance in data; and then choose these
principle components for our further analysis. So in this particular case, we identify three
principle components using these three principle components we fit a relationship between the
rainfall in a particular month. This is a monthly data that we are talking about, rainfall
in a particular month with the principle components.
So, we write this simple expression, this is a regression equation rainfall in the month
t is equal to some constant, and these are the k principle components, gamma k into p
c k t, so k is equal to 1 to k. Remember here, the principle components will be different
for different time periods. So, in this particular case, we have chosen three principal components
p c 1, p c 2, p c 3 all of which are different for different time periods, and these are
the gamma values. So, we write the rain t as a simple regression
form in a simple regression form as follows, when we do this for the training period; that
means, whatever data that we have used let say, between 1950 and 2003 we fit the regression
relationship and you get a R value of 0.789; which means, predicted and the observed the
coefficient between the correlation between predicted and observed is 0.789, which is
quite acceptable. In fact, we do several things to improve this,
let say we do clustering and then, we do fuzzy clustering and then, we add a seasonality
term etcetera, all those details we will not worry about. We will say that, we start with
a correlation of 0.789 between the observed and the predictor predicted. When we go into
the future, now this is this equation is fit using the historical data, these pc’s that
are obtained are using the historical data. We use the same principle directions into
the future and then, use this equation for projected values of the climate variable,
which is the mean sea level pressure in this particular case.
When we fit the relationship as I explained in our multiple linear regressions, the residuals
have to be normal, they have to follow normal distribution and therefore, we check for the
normality of residuals. What do I mean by residuals? That is, the predicted minus the
observed that error, that error series is called as a residual series. So, we fit the
residual series, this is a unstandardized non standardized residuals. So, as they are
obtained we fit it and then see that, it more or less fits a normal distribution.
In fact, there is more rigorous test that we carry out to examine whether the residuals
in fact, are normal and they are uncorrelated and so on; which I have explained in time
series analysis course, all of time series analysis topics, all of those test can be
carried out. So, this is just to demonstrate that they in fact fit the normal distribution,
reasonably well.
Then, we also look at this is for the training of the model; that means, when we are building
the model, we look at how well the observed mean and the predicted mean agree with each
other, this we do for the wet season as well as, the dry season. The wet season is our
monsoon period June, July, August, September and the dry season is the remaining period,
so they fit very well in fact, 281.9, 283.3 and so on.
So, mean as well as median we check for the model and then, we also check with the Nash-Sutcliffe
coefficient, which is generally used in our hydrologic literature, which is given by,
this is a observed at time t, this is predicted at time t and this is the mean for the entire
time period. So, this is how we calculate the Nash-Sutcliffe coefficient and this is
0.83, which is very good.
Once we are satisfied that, the regression relationship that we have built is acceptable,
then what we do is? We go to this particular G C M and obtain the projections on the climate
predictor. In this particular case, we have used only one predictor, which is the mean
sea level pressure at those grid points look at this.
So, we are looking at the mean sea level pressure, provided by the G C M at these grid points.
Now these are available for download, if you go to the G C M site you will get the simulations
provided into the future for for all the grid points. So, you can extract specifying your
longitude as well as, latitude you specify and then, those grid points you will get the
mean sea level pressures directly from the G C M website. You use those mean sea level
pressures. In fact, there is a slight here as I mentioned, we may use reanalysis data;
and the reanalysis data, which is given by typically we use the NCEP and NCAR reanalysis
data.
And what happens is in most cases? Let say, this is a G C M grid point, your NCEP grid
point may be somewhere here, so this is NCEP grid point, this is a G C M grid point. Let
say, we are using a particular G C M and the G C M grid G C M is here, grid point is here,
whereas your reanalysis data is here. And we would have developed the reanalysis
we would have developed the regression relationship with the reanalysis data and therefore, when
we use the projections this projected variable or the projected simulations must be brought
to this grid point that is the NCEP grid point. So, we need to interpolate the values that
are provided by the G C M at particular locations to the NCEP grid points, and this we do typically
by what is called as a spherical interpolation. So, we use spherical interpolation for interpolating between G C M grid points
and the NCEP grid points, this is just a broad idea, but there are lots of improvements that
are available that are possible in what I just told. And spherical interpolation is
readily doable using matlab programs. So, in matlab you have a routine to carryout spherical
interpolation you can use that toolbox and then in fact, there is a function for spherical
interpolation you can use that and then, do the spherical interpolation. But what is important
to understand here is that, you are developing these kinds of relationships with respect
to the reanalysis data. And the reanalysis grid points for example,
NCEP grid point, NCEP is National Center for Environmental Prediction U S A, those grid
points and the G C M grid points may not tally and therefore, you have to do an interpolation.
So, when we do this using the particular scenario that we are interested in as I said, we are
using B 2 scenario here; and with one particular model this is a C C S R N I E S model, this
is a Japanese model, we use this for the particular scenario; and then, project it into the future
upto 2003 or something we use it for building the model and then, the remaining part we
use it for projecting and this is a long term mean.
So, you may see certain trends that, the precipitation mean may be falling up to certain point and
then rising beyond a certain point as in time and so on. So, we see certain patterns as
we project it into the future, this is for the wet scenario, this is for the dry scenario,
dry scenario may be falling down like this. However, the point that is important and the
point, where the uncertainty is start arising is as you change this model and as you change
this scenario, you will get different pictures all together. So, instead of this model, if
you use another model, you get a different picture altogether, instead of this scenario
you choose another scenario let say instead of B 2, you may choose A 1 then you may get
a different scenario altogether. And that is what causes uncertainty, because you are
interested in obtaining the hydrologic scenario. For example, you are interested in seeing
how the precipitation is likely to change in future. And this uncertainty needs to be
addressed, if you are using these kinds of scenarios for our planning purposes, which
which is what is a final goal.
So, we convert this precipitation time series into what is called as drought indices. So,
we use the several drought indices are available, I will not going to details of those. We will
use what is called as a Standardized Precipitation Index S P I, which is a indicator of meteorological
drought, it is just require the precipitation values, using the precipitation values you
convert that into a standardized precipitation index. There are other indices like Palmer
drought severity index, Bhalme-Mooley index and then, effective drought index and so on.
But, the S P I, is the simplest one that we use for assessing meteorologic drought.
We convert the precipitation series into the S P I as follows; we fit a probability distribution
to the precipitation series. So, this is the annual rainfall here and then, we fit a probability
distribution C D F, using our Weibull method or whatever. So, these are Weibull’s plotting
position and typically annual rainfall as I have mentioned when I covered the probability
distributions, you may fit a gamma distribution. So, the gamma distribution in this particular
case fits reasonably good, then we convert this distribution to a standard normal distribution.
So, any point here you just transfer it into standard normal distribution and that gives
you the standardize precipitation value. What I mean by that is? Let say that, you
start with a particular value here go up, reach the curve and come horizontally reach
this point, and this value will be the S P I 12, that is how you convert the precipitation
time series into the S P I 12 values associated with various values of this. So, you get a
S P I 12 on the x axis here, associated with the standardized normal C D F, starting with
the C D F that you have fit for the annual rainfall. So, you get various values of the
S P I 12.
Now, this S P I 12 depending on the value that you get here, this will indicate the
drought category. So, 0 to minus 0.99, it indicates a normal situation and so on, mild
to moderate drought then, severe drought, and extreme drought etcetera. This is based
on the S P I that is standardize precipitation index and the reference is available here
Mckee et al 1993.
We use I just explain how we obtain the S P I 12 computation, so we do this both for
the observed rainfall as well as for the G C M output. So, with the G C M output we have
done the statistical downscaling using the regression relationship and then, we have
obtained into the future, we have obtained the time series of the precipitation and then,
we use we compute the S P I 12 for the future.
And then, look at how the S P I 12 is likely to evolve into future, when I use several
G C M’s different G C M’s for different scenarios? So, these are the different G C
M’s that we use for different scenarios. Remember, what I talked about just now was
precipitation projection. So, this gives the precipitation projection for a given G C M
and for a given scenario, this precipitation projection will then be converted into a S
P I projection, Standardized Precipitation Index to indicate how the droughts are drought
picture is likely to evolve in future.
. And that we do for several G C M scenario
combinations. So, each of them will show 1 G C M scenario combination. For example, this
curve may indicate A O M NASA for scenario, then A G C M C C S R for A 1 scenario, C G
C M 2 i s 92 a scenario. So, each curve represents the projection provided by a particular G
C M for a given scenario. As you can see here, there is a large amount of uncertainty associated
with these projections, even when you are looking at time periods of 2040’s.
So, there is a large spread that is being seen here, but this band or this spread starts
increasing as you go into the future 2040, to 2060 the band is much higher as you go
into 2080, it is much higher and so on. So, the uncertainty propagates into the future
and it is important for us to quantify the uncertainty when we want to use these projections
for our decisions as well as, develop methodologies and tools to see whether we can reduce the
band of uncertainty; because, we know with this kind of uncertainty, it is very difficult
for us to use this information into any decision making mechanisms.
One way of doing this is, let me show the same thing we are using the box plots. So,
when we have these kinds of projections, we generally provide as I told in the last class,
we provide the box plots. So, in 2020 we get a certain type of uncertainty, certain type
of range, and certain type of mean median and so on. 2040’s we show something, 2060
we show. So, typically the box plots are used to show the uncertainties as you progress
in time in the case of projections, using the climate change alright. Now, we look at
this, so at various points here 2020’s you have a certain ensemble of time series, so
this is essentially an ensemble of time series. So, at 2020’s you have certain spread and
certain realizations, and 2040’s you have certain different spread here and so on. If
you assume that, all this models and all the scenarios are equiprobable; that means, in
the future they all have the equal likelihood of providing the right projection; that means,
this projection is as likely to occur as this projection, because of our lack of knowledge.
We address the first level of uncertainty assuming that, all these projections are equiprobable
in future and then, fit probability distributions for various locations.
So, you look at 2020’s you fit a probability distribution based on the sample that is available
at 2020, 2040 you fit another probability distribution based on the
sample that is available at 2040 and so on.
So, you provide the probability distributions of the S P I 12 as the time progresses, this
may be 2020, this may be 2040, this may be 2060 and so on. We will then, examine how
the probability distributions are likely to change in future, the probability distribution
has absorbed or has used all the information that is available through these projections.
So, at any particular given time, it uses the several time series values that are possible
2040’s to 2060’s let say, this particular time frame, you will use all the time series
and then develop the models. So, these are typically done for time windows, not at a
particular time 2020, it may represent 2020 to 2040, similarly 2040 to 2060. So, use the
ensemble analysis and develop probability distributions at those time windows.
The way we do is, simply assume that every time window it follows a normal distribution
and then, get the parameters associated with that, associated with the normal distribution.
So, this way you are trying to address the uncertainty, first level of uncertainty you
simply say that, all models scenario combinations are equally possible. What I mean by that
is the, projection provided by any model scenario combination is as likely to be possible as
the projections provided by any other model scenario combination.
And then, you look at different time windows 2020, 2040. 2060 etcetera, you have an ensemble
of time series during that particular time period; assume that, this follows normal distribution
that means when you fit a distribution to that particular ensemble it follows a normal
distribution. And then, you estimate the parameters of that and then provide how the droughts
are likely to evolve in future. The other one will be you use a Kernel density
estimation, which I will explain presently and then, come out with the actual probability
distribution using the Kernel density function estimates. So, when you use the normal probability
distribution, you get such picture in this particular case, the near normal condition
the probabilities will be like this, the mild draught the probabilities will be like this,
extreme draught the probabilities are like this and so on. So, these are obtained from
assumption of normal distribution to the various time periods. So, this is 2000 to 2010, 2040
to 2050, 2090 to 2100 and so on. So, like this different time periods you use and estimate
the probabilities associated with the mild drought, extreme droughts, severe drought
and so on.
In the Kernel density estimation, what we do is that, we fit the probability density,
this is the Kernel density estimator of probability density, this is the number of observations
in our case, it will be the number of simulation that are possible, and this is called as a
smoothing parameter and you have the Kernel functions here. Now, this smoothing parameter,
which is called as a bandwidth, it is important to assume it is important to assume a certain
form of the bandwidth. And that in this particular case, we assume
that the Kernel is Gaussian and therefore, we get this kind of bandwidth here, we will
not go too much into detail. We just distinguish this between what we get with a normal density
as well as, Kernel density estimator.
The kernel density estimator again can be obtained directly from a matlab subroutine.
So, when you do the Kernel density estimator, you get near normal condition like this, and
then the severe drought extreme drought probabilities are obtained like this. So, essentially then,
what we are doing is, that we are using the downscaling technique to project into the
future in this particular case study, I have talked about projection of the precipitation
at a meteorological subdivision and specifically the Orissa subdivision.
We project the precipitation into the future, using several G C M scenario combinations;
convert the precipitation into a drought index. And in this particular case, we have used
S P I 12 the Standardized Precipitation Index, which is computed based on the monthly data
and that is why 12 months data comes into picture, because you get a spread, because
you get an uncertainty band at various time windows 2020 to 2040, 2040 to 2060 etcetera,
your fit probability distributions of the S P I 12 for those windows.
The first level of assumption that you make is, that you simply assume that at each of
these time windows, it follows normal distribution and then, obtained the parameters of the normal
distribution namely the mean and the standard deviation, for those different time windows.
And then, assess what are the probabilities of various levels of drought extreme drought,
normal drought normal situation and moderate drought and so on, based on the ranges that
are available for the S P I 12. Next, you use a Kernel density estimator,
which can also be done using a matlab program and then, you again assess probability of
various levels of droughts. So, this is how you address the uncertainties arising out
of the G C M and the scenarios. As I mentioned, our primary hypothesis here is that the projection
provided by any G C M is as likely to occur as projection any G C M scenario combination
is as likely to happen as projection provided by any other G C M scenario combination; which
means, the we were saying that, all this time series are all these projected time series
are equally likely to occur in future based on that we address the uncertainty.
There are other other questions to be asked in the uncertainty, before I close I will
just explain those things and then we close the lecture. It is possible that, these are
all not likely to these are not equally likely then, what we do? We start assigning weights
to the projection provided by a certain G C M scenario combination.
How do we assign the weights? We assign the weights based on, how well this G C M scenario
combination has performed for the particular region in the immediate past let say, between
1990 and 2005 or 2010, 1990 and 2010 how the particular G C M scenario combination is able
to reproduce exactly what has happen to the precipitation in that Orissa subdivision between
1990 and 2010 based on how well it has been able to reproduce we assign weights.
So, different G C M scenario combinations we will have different weights, and we provide
what are called as the weighted probability distributions. Then we also address a question
of the imprecision in the probability assessment, because of the small sample as I said even
if you get all the models and all the scenarios, perhaps you may have about 100 samples, 100
time series. So, at any particular location when you are looking at the probability distribution
for a particular month, you have maximum of 100 values.
So, this is a small sample that we are talking about in fact, even 100 you will not have.
As I as I showed in the particular case, you may have about 2025 values, and 2025 values
you are trying to fit a probability distribution. Therefore, the probability distribution itself
will not be precise then, we move onto what are called as a imprecise probabilities. So,
we develop imprecise probability distributions and then, provide of the droughts, that is
what is the probability bound for extreme drought, what is the probability bound for
normal conditions and so on. Now, these information, this types of information
will be extremely useful in planning for the future. Let say that, you want to plan for
agricultural use of water onto the future, so you you must know what levels of probabilities
exist for drought in various regions of the country and these kind of information will
be useful for that. So, today’s was a special lecture on using the stochastic hydrology
techniques that we have learnt in the particular course, for the most timely problem of climate
change impact assessment. And as I mentioned in the class in this particular
lecture, the climate change impacts the assessment of climate change impacts is burdened with
a large amount of uncertainty arising out of the G C M uncertainty, the general circulation
model uncertainty, the scenario uncertainty, the downscaling uncertainty and the intra-model
uncertainty themselves that is a model uncertainty. So, it is important for us to use the techniques
and the tools that we have learnt in the course, to address all these kinds of uncertainty.
And I have provided a simple example of using the G C M outputs to project Orissa rainfall
into the future and then, addressing uncertainties by providing probability distributions for
different time windows. And these probability distributions will be useful for planning
purposes using the probability distributions you can plan for agriculture activities, you
can plan for you can convert them into stream flow scenarios and so on, and the plan for
reservoir operation. So, developing adopt your responses will require these kind of
projections. So, the next lecture, which is the lecture
number 40 will be the last lecture of the course, where I will summarize whatever we
have taught we have discussed during all this 39 lectures, thank you for your attention.