evolved and random perturbation methods for calculating
TRANSCRIPT
Evolved and Random Perturbation Methods for Calculating Model Sensitivities and Covariances
William J. Martin* Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, Oklahoma
Ming Xue
Center for Analysis and Prediction of Storms and School of Meteorology, University of Oklahoma, Norman, Oklahoma
Submitted to Monthly Weather Review December, 2006
Revised May, 2007
*Corresponding author address: Dr. William Martin
Center for Analysis and Prediction of Storms National Weather Center, Suite 2500
120 David L. Boren Blvd Norman, OK 73072.
Phone: (405) 325-0402 E-mail: [email protected]
2
ABSTRACT
Different ways of perturbing the initial condition of an ensemble of forecasts for the purpose
of calculating sensitivity or covariance fields of model variables are examined. The three
methods considered are: random perturbations at each grid-point, smoothed random
perturbations, and perturbations that are evolved by the model through time from an earlier set of
perturbations. A very large ensemble of model runs using spatially discrete perturbations is also
compared for validation purposes. An ensemble size of 2000 members is used so as to reduce
the noise in sensitivity fields. Covariances found from the three methods are highly accurate and
nearly identical for any perturbation method. The calculation of sensitivity fields, however, is
more dependent on the perturbation method. For the cases of evolved or smoothed perturbations,
the spatial correlation of the perturbations leads to an inherent smoothing of the sensitivity fields.
Sensitivity structures of scales smaller than the perturbation correlation distance can not be
found. This is a particular problem for the evolved perturbations in the boundary layer.
Furthermore, the spatial correlation of initial perturbations makes the calculation of sensitivity
values inaccurate unless the complicated problem of separating the combined effects of
correlated perturbations on the forecast is dealt with. Consequently, mathematically correct
sensitivity values are only found by using initial perturbation fields that are spatially completely
random.
3
1. Introduction
In recent years, ensemble forecasts have gained increasing importance. Small ensembles
of order 10 members are used to provide probabilistic forecast information and the forecast
uncertainty (Buizza 2000; Toth and Kalnay 1997); moderate-sized ensembles of order 10 to 100
members are being used with data assimilation for various forms of ensemble Kalman filtering
(EnKF) (Evensen 1994). Moderately sized to very large sized ensembles have also been used to
calculate sensitivities statistically in a manner analogous to the calculation of covariances in an
EnKF method (Beare et al. 2003; Hakim and Torn 2005; Martin and Xue 2006, 2007). Hamill et
al. (2003) have used ensembles with up to 1600 members for the approximate calculation of
singular vectors as part of an ensemble data assimilation system. In addition to its value in data
assimilation and forecasting, information derived from ensembles can potentially be used for the
targeting of observations (Bishop et al. 2001), for possible weather modification, and for
discovering physical connections between different processes in the atmospheric model.
The uncertainty in a forecast depends on the error in the analysis (for a perfect model). If
perturbations smaller than analysis errors are used (as is the case in this work), then the forecast
variance is expected to be smaller and not useful as an estimate of forecast skill, but covariance
and sensitivity values can still be meaningful by themselves. Indeed, the sensitivity gradients
found by an adjoint model are exactly those that would be found with infinitesimal perturbations.
In situations where the variance in the initial conditions of an ensemble is lower than analysis
errors, covariance inflation (Anderson and Anderson 1999) might be used in some cases to
empirically increase the covariance for some sets of perturbations for use in an EnKF (these are
4
then modified by the assimilation cycle of an EnKF). This is not the purpose of our current
study, however.
Initial perturbations are generally either random or Monte Carlo in form, where
perturbation fields are created using random numbers (Mullen and Baumhefner 1994); or are
physically based, where fields are derived from the model itself (singular vectors, Lyapunov
vectors, and bred vectors). Other methods include lagged average forecasting (Hoffman and
Kalnay 1983) in which an ensemble is formed from runs of the same model initialized at
different times; multimodel ensemble techniques (Krishnamurti et al. 2000), which form an
ensemble from different prediction models; and methods that involve perturbing the
observations, model parameters, and boundary conditions (Houtekamer et al. 1996a,b;
Houtekamer and Mitchell 1998).
Singular vectors correspond to the fastest growing error modes of a model (in a tangent-
linear framework), and there use as initial perturbations is thought to be optimal for the
generation of accurate forecast error covariances with the smallest possible ensemble size
(Ehrendorfer and Tribbia 1997). In order for the forecast ensemble to be unbiased, theoretically,
the selection of initial perturbations should also be unbiased, and therefore should be selected in
some appropriate manner from the distribution of possible analysis states (Ehrendorfer and
Tribbia 1997). A review of various methods of ensemble initialization, for the purpose of
ensemble forecasting, can be found in Houtekamer and Mitchell (1998).
Different from most studies that deal with the optimal ensemble perturbations for an
ensemble forecast or for ensemble-based data assimilation, where the mean and ensemble spread
are the most important, we examine in this paper different ways of initializing an ensemble for
the purpose of accurately calculating the forecast sensitivities and covariances. Here the
5
sensitivity is defined as the gradient of a model output quantity, usually a scalar, with respect to
individual elements in the model input or initial condition state vector (Martin and Xue 2006;
2007). Since the sensitivity is a gradient field, the magnitude of the initial perturbations of the
ensemble does not matter so long as the perturbations are large enough for the machine
truncation error to be insignificant and the perturbations do not cause unrealizable physical
behavior. In this case, it is the ability of the initial perturbations in delineating the structure
details of the sensitivity fields that is important. In the case of covariance calculation, the
magnitude of the initial ensemble will affect the variance, i.e., the diagonal components of the
covariance matrix, but to a much smaller extent, if at all, on the correlation structure (a
covariance matrix B can be written as 1/ 2 1/ 2≡B D CD where C is the symmetric correlation
matrix and D1/2 as diagonal matrix made up of the square root of the variances) (Kalnay 2002).
For the calculation of initial condition sensitivity fields, it would be ideal to perturb every
degree-of-freedom (DOF) of the model each in an independent model run, as was done by
Lorenz (1968) for a 28 parameter model. This is not currently practical because of the large
number of DOFs in a typical modern, atmospheric prediction model. Each field variable value at
each grid point constitutes a DOF. For the sample case considered here, there are six variables
(not counting microphysical variables) defined at each of 1 million grid points. The approach
taken by Martin and Xue (2006) was to reduce the number of DOFs to consider by defining
perturbations at a 2-D array of multiple grid-point patches. While highly accurate, this method
could only be applied to the problem of finding 2-D sensitivity fields, and still required over
2000 forward model runs. Martin and Xue (2007) approached this problem by perturbing every
DOF of the model simultaneously and randomly and then using statistics (linear regression
analysis) to determine the desired relations among variables. Adjoint models (Hall and Cacuci
6
1982, 1983; Errico 1992; 1997) have also been used for calculating initial condition sensitivity
fields. Such models, however, are difficult to implement and can not estimate nonlinear
sensitivity; though once developed, they are computationally more economical.
By any perturbation method, the perturbed initial conditions lead to perturbed forecasts.
These perturbed forecasts can be related to the perturbed initial conditions in various ways, as
described in section 4 below, to produce forecast sensitivities (as either partial derivatives or
time-lagged covariances), as well as covariances between variables at the same time (Martin and
Xue 2007).
In this paper, several methods of perturbing the model initial conditions for the purpose
of calculating covariances and sensitivities are compared, including the random method of
Martin and Xue (2007, referred to as method “RAND”), the smoothed Gaussian method of Tong
and Xue (2006, referred to as method “SGAU”), and using perturbations evolved by the model
itself from a previous ensemble, similar to the bred vector method (referred to as method
“EVOL”). Because initial condition sensitivity fields are relatively noisy when calculated by
these methods, we employ ensembles of 2000 members.
2. The ARPS model and case used for study
a. The ARPS model
The prediction model used in this study is the Advanced Regional Prediction System
(ARPS, Xue et al. 2000, 2001, 2003). The configuration of the ARPS model used for this study
is the same as that in Martin and Xue (2006, 2007), which used a 9 km horizontal resolution grid
and 135×135×53 points centered over the IHOP (International H2O Project) study area in the
southern Great Plains of the United States (Weckwerth et al. 2004). The lateral boundaries were
forced by the 1800 UTC 24 May, 2002 NCEP ETA model forecast on the 40 km grid. Further
7
details of the model configuration will not be repeated here while the model details can be found
in the afore-referenced model description papers.
A problem with using forced lateral boundary conditions with this study is that the lateral
boundary conditions are the same for all the perturbed model runs (unless a technique is
employed to perturbed the boundaries, which we have not implemented). This means that
forecast field values near the boundaries will be largely determined by the boundary values
(rather than perturbations), with a consequent low variance in the forecast near the boundaries.
Consequently, statistical inferences near the boundaries may be inaccurate. This issue is
discussed further in Martin and Xue (2007). In that paper, it is shown that the forecast fields
after 6 hours of integration in the interior of the domain within 100 km to 200 km of the lateral
boundaries have little variance. Most of the interior of the domain, however, is not affected.
b. Case used for study
The case used here is the same as that from Martin and Xue (2006). This was a
convective initiation case from the 2002 IHOP field program, studied at high-resolution by Xue
and Martin (2006a,b). In this case, a dryline-cold front triple-point was in place in the eastern
Texas panhandle at 1800 UTC, 24 May, 2002. This triple-point moved southward during the
afternoon, reaching just southwest of the southwest corner of Oklahoma by 0000 UTC, 25 May.
The observed convection initiated south of the triple-point along the dryline by 2100 UTC, 24
May. In the ARPS 9 km resolution simulation, convection also occurred along the cold front in
northwest Oklahoma and southern Kansas, though this was not observed. Figure 1a shows the
three hour forecast of water vapor and wind 10 m above the ground, valid at 2100 UTC, 24 May,
and Fig. 1b shows the six hour forecast of total accumulated precipitation, together with surface
winds. For most of the analyses in this paper, perturbations were applied at 2100 UTC and the
8
ensemble integrated for 3 hours. This is to facilitate the comparison with evolved perturbations
from forecasts that started at 1800 UTC, a time when initial perturbations were examined in
Martin and Xue (2006, 2007). Since the relatively coarse resolution used by the very large
ensembles and the absence of data assimilation cycles that can include high-resolution local data,
the model forecast is not as good as those of Xue and Martin (2006a,b). Getting an accurate
forecast is not the primary goal of this study, however, as long as the model prediction is
physically realistic.
3. Perturbation Methods
Four methods are used in this study for perturbing the initial conditions of an ensemble of
model runs. For convenience, these will be abbreviated RAND (random perturbations at
gridpoints), SGAU (smoothed Gaussian perturbations), EVOL (evolved perturbations), and VLE
(a very large ensemble of discrete perturbations as used in Martin and Xue 2006).
a. Random perturbations at model gridpoints (RAND)
This is the method used by Martin and Xue (2007). In this method, a uniformly
distributed, quasi-random number between 0 and 1 is calculated at each grid point and for each
variable that is to be perturbed. A perturbation that is a constant fraction of the unperturbed
variable is then added to the variable at each grid point. If the random number is greater than or
equal to 0.5, the perturbation is added, while if the value is less than 0.5, it is subtracted. These
perturbations are thus binary, with only two possibilities at each grid point. This is perhaps the
simplest method for perturbing a model and has the advantage of independently perturbing each
degree of freedom of the model. An alternative and even simpler method is to add or subtract a
fixed amount (rather than a percentage of the unperturbed variable value) to each variable at each
grid point depending on the random coin flip. This is problematic as a fixed magnitude
9
perturbation may be too large in some parts of the model domain and too small in others for
certain variables. For example, a perturbation of ±1 g kg-1 in water vapor is reasonable in the
planetary boundary layer, but is too large at upper levels where such perturbations can lead to
unrealistic behavior, such as the instant formation of super-saturated clouds. It is therefore
desirable to perturb model variables by an amount that is some small fraction of its normal
variance. Using perturbation amounts equal to a percentage of the unperturbed variable only
achieves this if the percentage magnitude is chosen from experience as the variance is not
directly reflected by the magnitude of the unperturbed variable. In our experience (Martin and
Xue 2007), perturbations of 2-10% seem to work well for water vapor perturbations, while 0.2%
seem to work well for potential temperature perturbations (in Kelvins), in that these perturbations
were found not to be so small as to be lost by round-off error nor too large to lead to obviously
unrealistic nonlinearities.
The idea of the RAND method is to determine the response to perturbing each degree of
freedom (DOF) of the model independently, while actually perturbing all the DOFs
simultaneously and randomly. The response of the model to a perturbation of a degree of
freedom is the signal being sought, even if this response is diffusion or model adjustment. The
fact that the random perturbations are rapidly affected by dissipation during the model
integration, does not change the fact that the net affect is the sum of all the individual DOF
perturbations. In this Monte Carlo approach, the model response to a perturbation of a particular
DOF is found by a statistical method in which the randomness of the perturbations in all the
other DOFs cancel-out. This is approximately the result that would be found by perturbing just
that one DOF.
10
b. Spatially smoothed, Gaussian random perturbations (SGAU)
This method is used by Tong and Xue (2007) to initialize their ensemble for EnKF data
assimilation. For each variable at each grid point with indices (l,m,n), a spatially smoothed
random Gaussian perturbation is added of an amount:
�∈
=Skji
kjiwkjirhnml),,(
),,(),,(),,(δ , (1)
where r(i,j,k) is a random number at the grid point (i,j,k) sampled independently from a Gaussian
distribution with zero mean and unit standard deviation; w(i,j,k) is a three-dimensional distance-
dependent weighting function; and h is a scaling parameter. The fifth order correlation function
of Gaspari and Cohn (1999, Eqn. (4.10)) is chosen for w. The summation is over all grid points
within a specified radius from (l,m,n), which is chosen to be 27 km here (chosen to be between
the length scale of the perturbations typical of the RAND and EVOL methods). The end result
of (1) is a spatially random field with spatial correlation scales related to the spatial scale of the
correlation function w. The scaling factor h is chosen to obtain a random field with a desirable
variance or standard deviation.
c. Evolved perturbations from an ensemble (EVOL)
For this method, an ensemble is numerically integrated from initial states that are
randomly perturbed (in this case by the RAND method above). This integration is carried up to
the time when perturbed initial conditions are desired. The difference between the model fields
of each member of the ensemble and the unperturbed model run then can function as
perturbations for use in subsequent perturbation runs. These evolved perturbations will tend to
have much larger spatial scales than either grid point perturbations or smoothed perturbations.
This is due to the diffusion in the model acting throughout the integration, serving to spread and
11
smooth initial grid-scale randomness (the model employed fourth-order advection in the
horizontal and vertical, a 1.5 order TKE based subgrid turbulence and PBL model, as well as a
small amount of fourth order numerical diffusion). Other physical processes, such as surface
insolation, evapotranspiration, or convection can also affect the nature of these perturbations.
This method will also remove any gross imbalances introduced into the model by random
perturbations. The length scale of the perturbations from this method will depend on the length
of time integration from the initial state and the manner in which the model was originally
perturbed. This is similar to the bred vector method of Toth and Kalnay (1997) except that the
evolved perturbations are not rescaled. Such perturbations are also similar to those found in the
EnKF data assimilation system cycles prior to scaling and the assimilation of new observations
during an analysis cycle. Unlike the SGAU and RAND methods, these perturbations are
potentially biased. They are also not necessarily Gaussian.
For using the EVOL method here, the model is initialized at 1800 UTC using the RAND
method and integrated for 3 hours producing a set of evolved perturbations. The ensemble is
then integrated another 3 hours using these evolved perturbations. The final forecasts at 0000
UTC are no different from the straight forecast from 1800 UTC starting with the initial random
perturbations; the difference is that the evolved perturbations at 2100 UTC are recorded and
therefore known.
d. A very large ensemble of model runs with discrete perturbations (VLE)
This method is examined in Martin and Xue (2006). A similar method was used by
Beare et al. (2003) who referred to it as “sensitivity mapping” . It is used in this paper for
comparison with the methods described above. It is a direct perturbation method and does not
make use of random perturbations. For this method, a specific perturbation is applied at one area
12
in the model domain and in one variable. For example, a perturbation in boundary layer
moisture might be made at one location in a patch 3x3 grid points in size. The model is then
integrated forward in time and the difference between the perturbed and unperturbed model runs
provides the exact impact of that perturbation on the forecast. By using a Very Large Ensemble
(VLE) in which each member of the ensemble has the discrete perturbation at a different
location, sensitivity fields can be derived as the model response to individual perturbations, if
these generally non-overlapping perturbations fill the 2-D domain for which the sensitivity map
is to be constructed.
4. Calculation of covariances and sensitivities
Sample perturbation fields from each of the four methods described in section 3 are
shown in Fig. 2. Figures 2a,b,c show a single realization of perturbed boundary layer water
vapor which is arrived at by adding together the perturbations from each of the 9 vertical levels
in the lowest 1 km of the model. In each subfigure, the 10 g kg-1 isopleth of moisture is drawn
for reference. This line is close to the location of the dryline and cold front. The perturbation
magnitudes in Figs. 2a,b,c are larger to the south and east of this line where humidity is higher,
because the perturbations are defined as a percentage of the unperturbed moisture field. The
VLE perturbation sample shown in Fig. 2d is a single perturbation 27 km by 27 km wide by 9
km deep. In principle, a separate model run is made with this perturbation at a different one of
the possible 2025 different non-overlapping locations required for this perturbation to tile the
entire 2-D horizontal domain.
It is noteworthy that the effective length scale of the random perturbation methods varies,
with the RAND method (Fig. 2a) being the smallest, and the EVOL method (Fig. 2c) being the
largest. For the SGAU method (Fig. 2b), the mean length scale is selectable, and was selected
13
here to be a 27 km radius. It is also noted that the perturbations along the boundary of the EVOL
method are much weaker than in the interior of the domain. This is due to the effect of the
boundary conditions. The boundaries of the model run are externally forced by a larger scale
model run (the ETA model) which was not perturbed. Finally, we note that the magnitude of the
EVOL perturbations is much smaller than either the RAND or SGAU perturbations. The EVOL
perturbations were generated by first perturbing the ensemble 3 hours prior to the time of Fig. 2c
using perturbations identical to those of the RAND method. After the 3 hours of model
integration, these RAND perturbations have spread in space by primarily advection and
diffusion, processes which have reduced the perturbation magnitudes by an order of magnitude.
Covariance and sensitivity values can both be derived from the same ensemble of runs
(Martin and Xue 2007). To find sensitivity values from an ensemble of M members, two sets of
scalars need to be defined: the response function values, Ji, which are the scalar forecast
quantities of interest (one value from each member of the ensemble, with i being the index for
ensemble members); and the perturbation quantities of an initial DOF of the model, 0
ilx∆ (also
one value for each member of the ensemble). The subscript l refers to one element of the model
state vector,x , and the superscript 0 refers to the initial time. The statistical relation between
these M ordered-pairs of values can then be found by either a covariance sum or by statistical
linear regression. Sensitivity fields can be found simply by considering each component of the
vector or field individually.
Figures 3a, b, and c show sample scatter plots for which the statistical relation might be
sought. For the example of Fig. 3, the response function is the total rain which fell in the three
hour forecast (from 2100 UTC to 0000 UTC) in the box drawn southwest of Oklahoma in Fig.
6a. The perturbation quantity is the amount the boundary layer moisture was initially perturbed
14
at a particular grid location we knew to have a strong relation to the selected response function
(which was at the local correlation field maximum in Fig. 6a).
The banded structure of Fig. 3a is an artifact of using a binary coin-flip for choosing
either a positive or negative perturbation at each grid point. The boundary layer perturbations
are arrived at by averaging the perturbation at each of the lower 9 model levels together. If these
levels had the exact same value of humidity, then there would be only 10 possible values of the
perturbation, depending on the 9 coin flips; with the least likely being either all positive or all
negative perturbations. The breadth of the vertical lines is caused by the fact that all of the 9
levels do not have the same humidity, though they are close, as these points are in the (well-
mixed) convective boundary layer at 2100 UTC.
Since Figs. 3a, b, and c are all plotted on the same scale, it can directly be seen that the
sensitivities implied by these plots differ with perturbation method, with the EVOL method
having the steepest slope, and the RAND method the shallowest. Because sensitivity is a
dimensional physical quantity, it might have been expected that all three of these relations would
have been the same, in the absence of noise. The reason they are different will be discussed in
section 5.
a. Covariance and correlation coefficient
As discussed in more detail in Martin and Xue (2007), the covariance between a forecast
scalar quantity, J, and the forecast state vector (which includes all the model fields) at time t,�� ��
x ,
is calculated for each element, tlx , of
�� ��
x from an ensemble randomly different model runs of size
M as:
�=
−−=M
i
tl
tliJx
xxJJM i
tl
1
))((1σ . (2)
15
J itself can be defined as any scalar function of�� ��
x . The non-dimensional form of covariance is
the correlation coefficient, � :
tl
tl
xl
xJ
Jx
Jx σσσ
ρ = . (3)
In Fig. 4a,b,c we plot the correlation coefficient between the response function, J, defined
as the total rain which fell along the dryline between 2100 UTC and 0000 UTC, and the field of
10-m potential temperature at 0000 UTC. This example was chosen because it has a fairly
complex correlation structure, with a negative local minimum in the center, surrounded by a ring
of positive correlation, surrounded by areas of both positive and negative correlation in a
complex pattern. From all three perturbation methods, subtle details of the correlation structure
have been almost identically reproduced. From Fig. 1b it is seen that precipitation fell in and
around the central minimum in the box drawn in Figs 4a,b,c. We interpret the negative
correlation between this rainfall and the temperature field at the forecast time as being due to
evaporative cooling associated with the rainfall and downdraft. The ring of positive correlation
around this central minimum is likely related to positive correction that should exist between the
rainfall amount and the boundary layer moisture in the environment not directly affected by the
downdraft. Other areas of positive and negative correlation around this ring are more difficult to
interpret and may be related to gravity wave oscillations triggered by the convection; they are
nonetheless well-defined. For each of these ensembles, the ensemble size, M, was 2000, which
is apparently more than needed for finding accurate covariance fields between variables at the
same forecast time.
For the fields of Fig. 4, a signal-to-noise ratio (SNR) has been calculated. The signal is
defined as the peak magnitude of the sensitivity field, and the noise level is defined as the root-
16
mean-square value of the sensitivity field at all locations 20 or more grid points (180 km) away
from the center of the response function box. The SNR in dB is then:
levelnoise
signalSNR 10log10= . (4)
The SNR of the sensitivity fields of Figs. 4a,b, and c are practically identical, being 9.9,
10.0, and 10.1 dB, respectively. The slightly higher noise level from the EVOL method is
possibly due to an increase in round-off error from the smaller perturbation magnitudes. Clearly,
the method of perturbation has made little difference so far as the calculation of the correlation
between variables at the forecast time is concerned, and this was true for many other fields we
examined but do not show.
b. Sensitivity
Sensitivity can be defined in terms of either a partial derivative (a gradient) or a
sensitivity covariance. As discussed in Martin and Xue (2007), the two are closely related.
When defined as a covariance or correlation coefficient, sensitivity is calculated by simply
replacing�� ��
x with the state vector at the initial time,�� ��
x , in (2) and (3) above.
Figure 5a plots the correlation coefficient between the forecast (at 0000 UTC) boundary
layer water vapor in the small box drawn south of the Oklahoma-Texas border (which is 3x3 grid
cells in size), and the field of initial (2100 UTC) boundary layer water vapor, for the RAND
case. This region of the domain did not have precipitation and the sensitivity is due largely to
advection and diffusion effects (Martin and Xue 2006). Because the initial fields were perturbed
randomly at each grid point, Fig. 5a exhibits grid point noise. Consequently, it is desirable to
apply smoothing to the sensitivity field. Figure 5b is the same field as Fig. 5a after a 2-D nine-
point smoother has been applied three times in the horizontal. This is to be compared with the
same field derived from the ensemble using the SGAU perturbations, shown in Fig. 5c, and the
17
same field derived from the ensemble using EVOL perturbations (Fig. 5d), both unsmoothed.
For the calculation of these sensitivity correlation coefficients, the ensemble size of 2000 was
necessary to reduce the noise to a tolerable level, though there remains a significant level of
noise. Because the noise level can be controlled by smoothing and the different methods used
different amounts of smoothing (explicit or implicit), the SNR is not a useful a measure. By
comparing Fig. 5a with Fig 2a, Fig. 5c with Fig. 2b, and Fig. 5d with Fig. 2c, it is clear that the
spatial scale of the noise scales directly with the spatial scale of the initial perturbations.
Smoothing of the result from method RAND increases the length scale of the noise and produces
a result (Fig. 5b) which has a noise level comparable to that of the SGAU and EVOL methods.
All three methods produce well-defined central maxima as would be expected from the effects of
advection and diffusion. One difference is the value of maximum correlation coefficient, which
is 0.56 for the SGAU method, 0.95 for the EVOL method, and 0.19 for the RAND method
(before smoothing). This is due to the reduced amount of uncorrelated noise in the EVOL
method relative to the other two; a result which might have been anticipated from the scatter
plots of Fig. 3, which shows that the sample EVOL scatter plot has much less noise than the
other two methods.
Another difference is the size of the central maximum. The EVOL method has produced
a region of sensitivity that is physically larger than either the SGAU or RAND methods. This is
due to the larger length scales of the initial perturbations. Initial perturbations at individual grid
points for the EVOL method (Fig. 2c) are correlated with perturbations at other grid points
within approximately 100 km, as compared with 27 km for the SGAU method and no correlation
for the RAND method. This spatial correlation of initial perturbations leads to a blurring of the
calculated sensitivity correlation coefficients. The spatially uncorrelated perturbations from the
18
RAND method allow for the greatest possible spatial precision (or resolution) in calculated
sensitivity fields, though the need for smoothing reduces this precision. There is a direct trade-
off between the smoothness of the sensitivity field and its spatial noise.
The importance of spatial precision and the value of the RAND method are further
illustrated by sensitivities in vertical cross section presented in Fig. 6. Figure 6a plots the 3-hour
sensitivity correlation coefficient between the total rain which fell in the box drawn, from 2100
UTC to 0000 UTC, to initial (2100 UTC) 10-m (not boundary layer) water vapor perturbations as
calculated from the RAND method. As expected, this shows a central maximum near the rainfall
maximum along the dryline. Figures 6b,c,d show vertical cross-section plots of the same
sensitivity field with the cross-sections taken along the line A-B drawn in Fig. 6a. The vertical
cross-section plots do show significant differences, with only the RAND method producing a
sensitivity field which shows significant vertical structure. The vertical length scale of the
perturbations for the SGAU method is selectable and was chosen in this case to be larger than the
depth of the boundary-layer. Consequently, no vertical resolution in sensitivities can by found.
Boundary-layer vertical mixing produces a similar effect for the EVOL method. This mixing
vertically homogenizes the boundary layer perturbations. Only the RAND method, in which the
perturbations are completely uncorrelated in the vertical, is capable of discerning the vertical
structure of the sensitivity pattern.
It is interesting that the maximum in sensitivity correlation coefficient shown by Fig. 6c
is approximately 1 km above the surface. It might have been anticipated that the maximum in
sensitivity would be near the surface where the moisture advection is greatest. One possible
explanation for this is that moisture near the capping inversion might be relatively important in
the process of breaking the cap for convective initiation.
19
5. Forecast sensitivity analysis from correlated perturbation fields
In Fig. 3 above, it was noted that the actual dimensional sensitivity values (the best fit
slopes for Fig. 3) arrived at by a regression of a response function against a perturbation differed
by perturbation method. To show this in detail, Fig. 7 is presented. Figures 7a,b,c show the
gradient (partial derivative) sensitivity of the boundary layer moisture in the small box indicated
in each figure at the forecast time to the initial boundary-layer moisture field as calculated from
each of the three methods. The gradient sensitivity is calculated as the least-squares regression
slope between the cost function and the initial grid point perturbations (the best fit straight-line
slopes of scatter plots like Fig. 3). This approximates the partial derivatives:
>∆∆<≈
∂∂
00ll x
J
x
J (5)
where the angle brackets denote the expected value. To make the comparison as direct as
possible, all fields in Fig. 7 were smoothed by 3 passes through a 9-point filter. For each
subfigure of Fig. 7, there is an increase in noise on the dry side of the dryline. This effect
disappears when such plots are non-dimensionalized (as in Fig. 5). This is caused by the fact
that the regression is inaccurate in areas with no (or weak) signal. As non-dimensionalization of
(5) involves multiplication by the standard deviation of 0lx , non-dimensionalization eliminates
this effect where the standard deviation of the perturbations is small, which is the case west of
the dryline.
From Fig. 7, we find that the dimensional sensitivity is an order of magnitude larger from
the EVOL method (Fig. 7c) than it is from the RAND method (Fig. 7a), with the SGAU method
(Fig. 7b) in between. The reason for this is that (5) is only correct for the RAND method. More
20
generally, each of the iJ∆ values from the ensemble represents to a first order approximation the
total differential of J, which by the chain rule is:
0xdJdJJ •∇=≈∆ �� ���� �� , (6)
so that each iJ∆ value depends on every partial derivative of J times the perturbation in each
DOF of the model. If the perturbations of each DOF are uncorrelated (as for the RAND
method), then
><∂∂=>< 0
0 ll
dxx
JdJ . (7)
When dJ is regressed against a particular 0ldx , (5) is recovered. If, on the other hand, 0
ldx is
correlated with other members of dx0 (as for the EVOL and SGAU methods), then dJ is larger
(for positive correlations) than that given by (7) and any results calculated from (5) will be larger
than the desired sensitivity gradient. Because the effects of all the correlated perturbations are
added together in (6), it is not a simple matter to separate them so as to obtain the sensitivity
gradient at a specific grid point. (6) represents a set of simultaneous equations which in principle
can be solved for the components of J�� ���� ��∇ . In practice, this solution is a difficult and possibly
ill-conditioned inversion problem.
For the EVOL method, perturbations at a point are correlated with nearby perturbations.
The spatial scale for this correlation would be expected to increase with the integration time used
for the production of the evolved perturbations because of diffusion effects. Consequently, the
longer the evolution time, the steeper the slope of a plot like Fig. 3c would be.
To explore the accuracy of the RAND technique and to provide some validation for it,
Fig. 8 is presented that compares the dimensional sensitivity derived from the RAND method
(Fig. 8a) with that from the VLE method (Fig. 8b). Both sensitivity fields were smoothed by 2
21
passes through a 9-point smoother. For Fig. 8, the sensitivity is between the forecast 10-meter
water vapor in the box drawn to the initial field of boundary layer moisture. For the VLE
method, the perturbations are 1-km deep columns of moisture at the surface 27X27 km in
horizontal size. In order to accomplish this direct comparison, these same perturbations are
synthesized for the RAND method by adding together the initial grid point perturbations in this
27X27X1 km region. These perturbations are then regressed against the Ji values as before. The
maximum from the RAND method is found to be 0.017, while that from the VLE method is
0.014, a difference of about 20%. It is possible that this difference is due to nonlinearity because
the perturbations from the VLE method are effectively much larger in magnitude than those from
the RAND method. It is also possible that the RAND and/or VLE result is contaminated by the
sweeping of unperturbed boundary information, as the region of sensitivity is less than 200 km
from the southern boundary.
6. Summary and Discussion
This study compared the covariance between forecast scalars and fields at a forecast time,
and compared the sensitivity of forecast scalars to initial fields, as calculated from ensemble
forecasts starting from initial conditions perturbed with three different random or quasi-random
methods. The ensemble size was very large at 2000 members, which was necessary because of
the need to reduce the noise in sensitivity fields. The three methods included random,
uncorrelated perturbations at each model grid point (RAND); smoothed Gaussian, random
perturbation fields (SGAU); and perturbations evolved from a previous ensemble of randomly
perturbed runs (EVOL).
For the calculation of the covariance between variables at the same forecast time, the
initial perturbation method did not make any material difference, despite large differences in the
22
spatial scale and magnitude of the perturbations. Of the methods used, the RAND method was
the simplest to apply. This similarity of covariance for the different perturbation methods may
be due partly to the fact that the perturbations were relatively small. From Fig. 2, the
perturbations of the EVOL method were typically 0.05 g kg-1. Those of the SGAU and RAND
methods were larger, but more localized. The effective perturbation magnitude of large length
scales from these latter two methods would be reduced due to cancellation of the small randomly
perturbed regions. If small perturbations are used, then the model response is more likely to be
linear, and the response to different kinds of perturbations more similar. Using perturbations
larger in magnitude for the SGAU or RAND methods is problematic because strong local, non-
linear pathologies occur (which are not realized in real cases). Also, using effectively small
perturbations may be undesirable when implementing and EnKF because the model variance is
combined with measurement uncertainty to produce the Kalman gain, so that an EnKF would
underweight observations as the forecast variance is related to the initial condition variance.
However, it might be possible to either inflate the model variance or deflate the measurement
uncertainty, if an ensemble based on small perturbations were used. One advantage of small
perturbations is that results for sensitivity similar to an adjoint are achieved, without the
difficulty of implementing an adjoint.
The EVOL method, surprisingly, had several shortcomings for use in sensitivity
calculations. Because the length scale of the EVOL perturbations was not controlled, some fine
details of sensitivity fields can not be found. For example, the sensitivity of the forecast to the
vertical moisture profile of the initial boundary layer could not be found for the case explored
here because the evolved perturbations did not have sufficient vertical structure in the convective
boundary layer due to turbulent mixing. Evolved perturbations are also affected by the lateral
23
boundary conditions. If the boundary conditions are not perturbed in some manner as was the
case here, then the evolved perturbations will not have structure near the boundaries. This would
affect results of both correlations between the forecast and the initial conditions, and between
different quantities at the forecast time, if these relations are desired near a lateral boundary.
There is an inherent trade-off between the smoothness of calculated sensitivity fields and
the spatial resolution. The RAND method inherently produces the best spatial resolution for
sensitivity fields, however the fields contain considerable grid scale noise. This noise can be
reduced by smoothing, but then some spatial resolution is lost. The SGAU and EVOL methods
have smooth and spatially correlated initial perturbations. This produces smoothing of the
calculated sensitivity fields as well.
Finally, both the SGAU and EVOL methods present particular difficulty in calculating
dimensional sensitivity values. Because the initial perturbations at each grid point are correlated
with perturbations at other grid points, it becomes a difficult inversion problem to derive
sensitivity fields. The RAND method does not have this problem because all the initial
perturbations are uncorrelated. The RAND method is the only method that can easily determine
sensitivity fields which are correct in magnitude
Acknowledgements
This work was primarily supported by NSF grants ATM-0129892 and ATM-0530814.
The second author was further supported by NSF grants EEC-0313747, ATM-0331594 and
ATM-0331756 and ATM-0608168. The computations were performed on the National Science
Foundation Terascale Computing System at the Pittsburgh Supercomputing Center. Thomas
Hamill, Robert Fovell, and an anonymous reviewer help to improve this paper.
24
REFERENCES
Anderson, J. L., and S. L. Anderson, 1999: A Monte Carlo implementation of the nonlinear
filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127,
2741-2758.
Bishop, C. H., B. J. Etherton, and S. J. Majumdar, 2001: Adaptive sampling with the ensemble
transform Kalman filter. Mon. Wea. Rev., 129, 420-36.
Beare, R. J., A. J. Thorpe, and A. A. White, 2003: The predictability of extratropical cyclones:
Nonlinear sensitivity to localized potential-vorticity perturbations. Quart. J. Roy. Meteor.
Soc., 129, 219-37.
Buizza, R., 2000: Skill and economic value of the ECMWF ensemble prediction system. Quart.
J. Roy. Meteor. Soc., 126, 649-68.
Evensen, G., 1994: Sequential data assimilation with a nonlinear quasigeostrophic model using
Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10143-62.
Ehrendorfer, M. and J. J. Tribbia, 1997: Optimal prediction of forecast error covariances through
singular vectors. J. Atmos. Sci., 54, 286-313.
Errico, R. M., 1997: What is an adjoint model? Bull. Amer. Meteor. Soc., 78, 2577-91.
Errico, R. M., 2003: The workshop on applications of adjoint models in dynamic meteorology.
Bull. Amer. Meteor. Soc., 84, 795-8.
Errico, R. M., and T. Vukisevic, 1992: Sensitivity analysis using an adjoint of the PSU-NCAR
mesoscale model. Mon. Wea. Rev, 120, 1644-60.
Gaspari, G and S. E. Cohn, 1999: Construction of correlation functions in two and three
dimensions. Quart. J. Roy. Meteor. Soc., 125, 723-57.
25
Hakim, G. J. and R. D. Torn, 2006: Ensemble Synoptic Analysis. Fred Sanders AMS
Monograph.
Hall, M. C. G. and D. G. Cacuci, 1982: Sensitivity analysis of a radiative-convective model by
the adjoint method. J. Atmos. Sci., 39, 2038-50.
Hall, M. C. G. and D. G. Cacuci, 1983: Physical interpretation of the adjoint functions for
sensitivity analysis of atmospheric models. J. Atmos. Sci., 40, 2537-46.
Hamill, T. M., C. Snyder, and J. S. Whitaker, 2003: Ensemble forecasts and the properties of
flow-dependent analysis error covariance singular vectors. Mon. Wea. Rev., 131, 1741-58.
Hamill, T. M. and J. S. Whitaker, 2005: Accounting for the error due to unresolved scales in
ensemble data assimilation: a comparison of different approaches. Mon. Wea. Rev., 133,
3132-47.
Hoffman, R. N., and E. Kalnay, 1983: Lagged average forecasting, an alternative to Monte
Carlo forecasting. Tellus, 35A, 100-18.
Houtekamer, P. L., L. Lefaivre, J. Derome, H. Ritchie, and H. L. Mitchell, 1996a: A system
simulation approach to ensemble prediction. Mon. Wea. Rev., 124, 1225-1242.
Houtekamer, P. L. L. Lefaivre, and J. Derome, 1996b: The RPN ensemble prediction system.
Proc. of the ECMWF Seminar on Predictability, Vol. 2, 121-146.
Houtekamer, P. L., and H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter
technique. Mon. Wea. Rev. 126, 796-811.
Kalnay, E., 2002: Atmospheric modeling, data assimilation, and predictability. Cambridge
University Press, 341 pp.
26
Krishnamurti, T. N., C. M. Kishtawal, Z. Zhang, T. LaRow, D. Bachiochi, E. Williford, S.
Gadgil, and S. Surendran, 2000: Multimodel ensemble forecasts for weather and seasonal
climate. J. Climate 13, 4196-216.
Lorenz, E. N., 1968: The predictability of a flow which possesses many scales of motion.
Tellus, 17, 321-33.
Martin, W. J. and M. Xue, 2006: Sensitivity Analysis of Convection of the 24 May 2002 IHOP
Case Using Very Large Ensembles. Mon. Wea. Rev., 134, 192-207.
Martin, W. J. and M. Xue, 2007: Determining sensitivities, impacts, and covariances from very
large ensembles with randomly perturbed initial conditions. Submitted to Mon. Wea. Rev.
Mullen, Steven L. and D. P. Baumhefner, 1994: Monte Carlo simulations of explosive
cyclogenesis. Mon. Wea. Rev., 122, 1548-67.
Tong, M, and M. Xue, 2007: Simultaneous estimation of microphysical parameters and
atmospheric state with radar data and ensemble square-root Kalman filter. Part I:
Sensitivity analysis and parameter identifiability Mon. Wea. Rev., Conditionally accepted.
Toth, Z. and E. Kalnay, 1997: Ensemble forecasting at NCEP: the breeding method. Mon. Wea.
Rev., 125, 3297-318.
Weckwerth, T. M., D. B. Parsons, S. E. Koch, J. A. Moore, M. A. LeMone, B. B. Demoz, C.
Flamant, B. Geerts, J. Wang, W. F. Feltz, 2004: An overview of the International H2O
Project (IHOP 2002) and some preliminary highlights. Bull. Amer. Meteor. Soc., 85, 253-
77.
Xue, M., K. K. Droegemeier, and V. Wong, 2000: The Advanced Regional Prediction System
(ARPS)-a multiscale nonhydrostatic atmospheric simulation and prediction tool. Part I:
Model dynamics and verification. Meteor. Atmos. Phys., 75, 161-93.
27
Xue, M., K. K. Droegemeier, V. Wong, A. Shapiro, K. Brewster, F. Carr, D. Weber, Y. Liu, and
D.-H. Wang, 2001: The Advanced Regional Prediction System (ARPS) – a multiscale
nonhydrostatic atmospheric simulation and prediction tool. Part II: Model physics and
applications. Meteor. Atmos. Phys., 76, 143-65.
Xue, M, D. Wang, J. Gao, K. Brewster and K. Droegemeir, 2003: The Advanced Regional
Prediction System (ARPS), storm-scale numerical weather prediction and data
assimilation. Meteor. Atmos. Phys., 82, 139-70.
Xue, M. and W. J. Martin, 2006a: A high-resolution modeling study of the 24 May 2002 case
during IHOP. Part I: Numerical simulation and general evolution of the dryline and
convection. Mon. Wea. Rev., 134, 149–171.
Xue, M. and W. J. Martin, 2006b: A high-resolution modeling study of the 24 May 2002 case
during IHOP. Part II: Horizontal convective rolls and convective initiation. Mon. Wea.
Rev., 134, 172–191.
28
List of figures
Fig. 1. Fields of (a) 10-m water vapor mixing ratio and wind vectors at 2100 UTC 24 May 2002
from a 3-hour forecast, and (b) total accumulated rainfall and 10-m wind vectors at 0000
UTC 25 May 2002 from a 6-hour forecast. Water vapor contour increment is 0.5 g kg-1
and rainfall contour increment is 10 mm. Length of 10.0 m s-1 wind vector is indicated at
the lower left corner of plots.
Fig. 2. Perturbed initial fields of boundary layer water vapor from four methods: (a) random
binary perturbations at each grid point, (b) spatially smoothed Gaussian perturbations, (c)
evolved perturbations, and (d) one example of a specific perturbation at one spatial
location. The 10 g kg-1 isopleth of moisture is drawn for reference. Contour increments
vary and are (a) 0.125 g kg-1, (b) 0.25 g kg-1, (c) 0.025 g kg-1, and (d) 0.1 g kg-1.
Fig. 3. Scatter plots of a response function versus initial perturbations at a point for three
methods of initial perturbation: (a) method RAND; (b) method SGAU; and (c) method
EVOL.
Fig. 4. Correlation coefficient between the total rain that fell in the rectangular box drawn in each
figure (from 21 UTC to 0 UTC) and the field of forecast 10-m potential temperature from
three different perturbation methods: (a) random grid-point, (b) smoothed Gaussian, and
(c) evolved. A portion of the political outline of Oklahoma is also drawn in the upper-
right corner of each figure. Contour increment is 0.1.
Fig. 5. Correlation coefficient fields between the total boundary layer water vapor in the drawn
box and the field of initial boundary layer water vapor perturbations, obtained for
different initial perturbation methods as indicated in the plots. Panel b shows a smoothed
29
version of panel a. Local maxima are (a) 0.195, (b) 0.143, (c) 0.558, and (d) 0.946.
Contour increments are 0.025 for (a), 0.0125 for (b), and 0.05 for (c) and (d).
Fig. 6. (a) Correlation coefficient between total rain which fell in the box drawn over 3 hours to
initial boundary-layer water vapor as found from method RAND. (b) same as (a), but in
vertical cross-section along the line A-B at y = 436.5 km. (c) same as (b), but from the
SGAU method. (d) same as (b), but from the EVOL method. Contour increments are
0.0125 for (a), 0.01 for (b) and 0.05 for (c) and (d). (a) and (b) have been smoothed by
two passes through a 9-point smoother. The 10 g kg-1 isopleth of 10-meter moisture has
been drawn in each figure for reference.
Fig. 7. Dimensional sensitivity of the boundary layer moisture in the small box indicated in each
figure at the 3-hr forecast time to the initial field of boundary-layer moisture
perturbations. From three perturbation methods: (a) random, (b) smoothed Gaussian, and
(c) evolved. Contour increments are, respectively, 0.1, 0.2, and 1.0 g kg-1 per g kg-1 for
(a), (b), and (c). The local maxima southeast of the response function box are: 0.42, 1.0,
and 5.4 for (a), (b), and (c). Fields for all subfigures have been smoothed by two passes
through a nine-point smoother.
Fig. 8. Dimensional sensitivity of 10-m moisture in small black box in each figure to initial field
of boundary layer moisture from (a) random initial perturbations at each grid point, and
(b) a very large ensemble of discrete perturbations. Contour increment was 0.0025 for (a)
and (b). Maximum in (a) is 0.017 and for (b) is 0.014.
30
0 200 400 600 800 10000
200
400
600
800
1000
(km)
(km
)
1010
3
34
4
4
4
5
5
5
6
6
6
6 7
7
8
8
8
9
9
9
10
10
10
11
11
11
12
12
12
13
13
13
13
13
13
14
14
14
0 200 400 600 800 10000
200
400
600
800
1000
(km)
(km
)
1010
20
60
a b
Fig. 1. Fields of (a) 10-m water vapor mixing ratio and wind vectors at 2100 UTC 24 May 2002 from a 3-hour forecast, and (b) total accumulated rainfall and 10-m wind vectors at 0000 UTC 25 May 2002 from a 6-hour forecast. Water vapor contour increment is 0.5 g kg-1 and rainfall contour increment is 10 mm. Length of 10.0 m s-1 wind vector is indicated at the lower left corner of plots.
31
10
1010
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
10
1010
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
(km
)
10
1010
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (
km)
10
1010
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
a
c d
b
Fig. 2. Perturbed initial fields of boundary layer water vapor from four methods: (a) random binary perturbations at each grid point, (b) spatially smoothed Gaussian perturbations, (c) evolved perturbations, and (d) one example of a specific perturbation at one spatial location. The 10 g kg-1 isopleth of moisture is drawn for reference. Contour increments vary and are (a) 0.125 g kg-1, (b) 0.25 g kg-1, (c) 0.025 g kg-1, and (d) 0.1 g kg-1.
32
500.
550.
600.
650.
700.
750.
800.
850.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
J, m
m r
ain
BL QV PERT, g/kg
500.
550.
600.
650.
700.
750.
800.
850.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
J, m
m r
ain
BL QV PERT, g/kg
500.
550.
600.
650.
700.
750.
800.
850.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
J, m
m r
ain
BL QV PERT, g/kg
a
b
c
Fig. 3. Scatter plots of a response function versus initial perturbations at a point for three methods of initial perturbation: (a) method RAND; (b) method SGAU; and (c) method EVOL.
33
-0.6
-0.6
-0.6
-0.4
-0.4
-0.4
-0.4
-0.2
-0.2
-0.2
-0.2
-0.2
-0.2
0.2
0.2
0.2
0.4
300 400 500 600 700300
400
500
600
700
x (km)
y (
km)
RAND SNR=9.9 db MIN = - 0.86
-0.8
-0.6
-0.6
-0.4-0.4
-0.4
-0.4
-0.2
-0.2
-0.2
-0.2
-0.2
-0.2
0.20.2
0.4
0.4
300 400 500 600 700300
400
500
600
700
x (km)
y (
km)
SGAU SNR=10.0 db MIN = - 0.86
-0.6
-0.6
-0.4
-0.4-0.4
-0.4
-0.4
-0.2-0.2
-0.2
-0.2
-0.2
-0.2
0.2
0.2 0.4
0.4
300 400 500 600 700300
400
500
600
700
x (km)
y (k
m)
EVOL SNR=10.1 db MIN = - 0.85
a
c
b
Fig. 4. Correlation coefficient between the total rain that fell in the rectangular box drawn in each figure (from 21 UTC to 0 UTC) and the field of forecast 10-m potential temperature from three different perturbation methods: (a) random grid-point, (b) smoothed Gaussian, and (c) evolved. A portion of the political outline of Oklahoma is also drawn in the upper-right corner of each figure. Contour increment is 0.1.
34
10
1010
0 200 400 600 800 10000
200
400
600
800
1000
y (k
m)
10
1010
0 200 400 600 800 10000
200
400
600
800
1000
y (k
m)
10
1010
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
10
1010
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
c d
baRAND RAND SMOOTH
SGAU EVOL
Fig. 5. Correlation coefficient fields between the total boundary layer water vapor in the drawn box and the field of initial boundary layer water vapor perturbations, obtained for different initial perturbation methods as indicated in the plots. Panel b shows a smoothed version of panel a. Local maxima are (a) 0.195, (b) 0.143, (c) 0.558, and (d) 0.946. Contour increments are 0.025 for (a), 0.0125 for (b), and 0.05 for (c) and (d).
35
10.0
10.0
10
.0
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
10.0
200 400 600 8000
1
2
3
4
5
x (km)
z (k
m)
0.0
0.0
0.0
0.1
10.0
10.0
200 400 600 8000
1
2
3
4
5
x (km)
z (k
m)
0.1
0.1 0.2
0.2
0.3
0.3
0.4 0.4 10.0
10.0
200 400 600 8000
1
2
3
4
5
x (km)
z (k
m)
0.1
0.1
0.2
0.2
dc
a b
AA B
RAND
SGAU EVOL
Fig. 6. (a) Correlation coefficient between total rain which fell in the box drawn over 3 hours to initial boundary-layer water vapor as found from method RAND. (b) same as (a), but in vertical cross-section along the line A-B at y = 436.5 km. (c) same as (b), but from the SGAU method. (d) same as (b), but from the EVOL method. Contour increments are 0.0125 for (a), 0.01 for (b) and 0.05 for (c) and (d). (a) and (b) have been smoothed by two passes through a 9-point smoother. The 10 g kg-1 isopleth of 10-meter moisture has been drawn in each figure for reference.
36
10
10
10
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (
km)
10
10
10
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
10
10
10
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
a
b
c
RAND MAX=0.42
SGAU MAX=1.03
EVOL MAX=5.42
Fig. 7. Dimensional sensitivity of the boundary layer moisture in the small box indicated in each figure at the 3-hr forecast time to the initial field of boundary-layer moisture perturbations. From three perturbation methods: (a) random, (b) smoothed Gaussian, and (c) evolved. Contour increments are, respectively, 0.1, 0.2, and 1.0 g kg-1 per g kg-1 for (a), (b), and (c). The local maxima southeast of the response function box are: 0.42, 1.0, and 5.4 for (a), (b), and (c). Fields for all subfigures have been smoothed by two passes through a nine-point smoother.
37
10
10
10
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
10
10
10
0 200 400 600 800 10000
200
400
600
800
1000
x (km)
y (k
m)
a b
Fig. 8. Dimensional sensitivity of 10-m moisture in small black box in each figure to initial field of boundary layer moisture from (a) random initial perturbations at each grid point, and (b) a very large ensemble of discrete perturbations. Contour increment was 0.0025 for (a) and (b). Maximum in (a) is 0.017 and for (b) is 0.014.