Title: Using GPS RO to determine the probability density of free tropospheric relative humidity
1Using GPS RO to determine the probability density
of free tropospheric relative humidity and
constrain how it is controlled E. R.
Kursinski1, S. Sherwood2, W. Read3 1University
of Arizona, 2Yale, 3JPL
GPS Conference August 2005
2Outline
- Motivation
- Accuracy of relative humidity from GPS RO
- Improving the RH estimates via deconvolution of
errors - Evaluation of simple RH distribution model
- Stochastic model explanation
- GPS MLS comparisons with model
- Single cell model
- Summary and conclusions
3Motivation for Moisture Observations
- Water is crucial to energy transport and
circulation within the Earth weather and climate
system through latent heat exchange - Precipitation largely controls the extent and
type of continental biosphere - Water vapor is the most important greenhouse gas
important throughout the troposphere and into the
stratosphere - Clouds strongly affect the radiation budget
through reflection scattering of shortwave
radiation and emission and absorption of IR - Water cools the surface in the form of clouds in
daytime, warms the surface through the greenhouse
effect as both a gas and as clouds and cools the
surface via evaporative cooling
4Motivation Evaluation of a Simple Model
- The tropics are where the magnitude and sign
water vapor feedback is generally believed to be
the most uncertain - We want as simple as possible an explanation of
how the water vapor distribution is controlled in
the tropics - Steve Sherwood has proposed a very simple model
- We evaluate it using GPS and MLS relative
humidity observations
5Testing a simple relative humidity model with
observations
- Model evaluation requires
- Relative humidity histograms to establish how
frequently each relative humidity range occur in
free troposphere - Global water vapor observations with good
vertical resolution independent of conditions
(especially clouds). - Inadequate observations
- Global analyses Unknown model humidity biases
- Radiosondes Poor spatial sampling, biases
- IR sounders Unknown biases associated with
clouds - Nadir passive microwave Inadequate vertical
resolution - Best available observations
- GPS RO Good for 2 to 9 km alt. in tropics
- MLS Good in upper troposphere, some sensitivity
to clouds
6GPS - ECMWF Specific Humidity Profile Comparisons
GPS (solid), ECMWF (dotted), saturation (dashed)
from July 2001 (g/kg)
7Subtropical Moisture Estimates
OLR
- GPS ECMWF comparison July 1995 10oS-25oS
Mean
Mode
8Deriving Relative Humidity from GPS RO
- Two basic approaches
- Direct method use N T profiles and hydrostatic
B.C. - Variational method use N, T q profiles and
hydrostatic B.C. with error covariances to update
estimates of T, q and P. - Direct Method
- Theoretically less accurate than variational
approach - Simple error model
- (largely) insensitive to NWP model humidity
errors - Variational Method
- Theoretically more accurate than simple method
because of inclusion of apriori moisture
information - Sensitive to unknown model humidity errors and
biases - Since we are evaluating a model we want water
vapor estimates as independent as possible from
models - gt We use the Direct Method
9Direct Method Solving for water vapor given N
T
(1)
- Use temperature from a global analysis
interpolated to the occultation location - To solve for P and Pw given N and T, use
constraints of hydrostatic equilibrium and ideal
gas laws and one boundary condition
Solve for P by combining the hydrostatic and
ideal gas laws and assuming temperature varies
linearly across each height interval, i
(2)
where z height, g gravitation
acceleration, m mean molecular mass of moist
air T temperature R universal gas constant
10Solving for water vapor given N T
- Given knowledge of T(h) and pressure at some
height for a boundary condition, then (1) and (2)
are solved iteratively as follows - 1) Assume Pw(h) 0 or 50 RH for a first guess
- 2) Estimate P(h) via
- 3) Use P(h) and T(h) in (1) to update Pw(h)
- 4) Repeat steps 2 and 3 until convergence.
11Solving for Water Vapor given N T
- To produce consistent statistics for the latitude
versus height histograms, we begin all of the
water vapor profiles in a given lat-hgt bin at
the same height determined as the height where
the average profile temperature first exceeds 240
K - Relative humidity U e/es(T) is calculated for
each profile with the saturation vapor pressure
over liquid used above freezing temperature and
over ice below the freezing temperature.
12Estimating GPS water vapor error
- The error in relative humidity, U, due to changes
in refractivity (N), temperature (T) and pressure
(P) from GPS is
where L is the latent heat and Bs a1TP / a2es.
- The temperature error is particularly small in
the tropics where we are focusing (1 - 1.25 K) - Refractivity error Based on simulations, the
refractivity error is approximated as
13Estimated GPS Relative Humidity Error (Tropics)
14Negative U and Error Deconvolution
- Problem The Direct Method U estimate can be
negative - gt produces an unphysical, negative tail in
the U histograms - Simplest correction is to push all negative U
values to the minimum positive U bin. - Systematically increases the mean (bad)
- Error deconvolution is theoretically better
approach - Model the measured Umeasured as Utrue eU
- Measured histogram (PDF) is convolution of the
true PDF and the error PDF - Such that PDFUmeas PDFUtrue PDFe
- IF we understand the error PDF we can in theory
deconvolve it from the measured PDF to recover
the true PDF
15U error deconvolution (work in progress)
- Represent the error convolution in matrix form, Y
A X - X is truth,
- A represents the convolution with the error PDF,
- Y is measured PDF
- Use negative tail to optimize the estimate of
error PDF - Assume PDFe is symmetrical
- Include 2 additional constraints
- Total Probability is conserved
- Mean of distribution is conserved (gt error has
zero mean) - Add these as entries in Y and rows in A
- Least squares problem because of entries in Y gt
X
16Error Deconvolution (contd)
- Finding the best PDFe is a trial and error
approach based upon which PDFe best matches the
observed - Found that a PDFe that is a linear combination of
gaussian and exponential works well - Summary
- Choose trial PDFe gt A
- (using lin. combination of gaussian
exponential) - Calculate least squares Xls solution
- Smooth Xls if necessary
- Forward calc YA Xls
- Choose A which minimizes the variance of Y-Y
- Approach yields both better estimate of X as well
as refined understanding and estimates of errors
17Deconvolution example
- Histogram of GPS relative humidity data
between 30oS and 20oS latitude and between 2 and
3 km altitude for July 2002. Dotted line is
histogram of measurements. Solid line is
sharpened histogram after deconvolving the errors.
18Data Sets for Moisture Variability Study
- 2000 occultations each from CHAMP in January,
April, July, October period - GPS canonical transform data smoothed to 200 m
vertically courtesy of Chi Ao at JPL - Interpolate the nearest ECMWF 12 hour, 22 level
global analysis to each occultation profile - Bin the data into a 2-D latitude vs. height grid
- Every 10 degrees in latitude
- Every 0.5 km in height (2 to 9 km altitude)
19Now on to the Model
20Stochastic Model Summary
- Parcels leaving a convective system possess some
initial relative humidity, R0, 100 established
by cloud physics. - leaving a convective system we define as
reaching a distance from the convective cores
where the advection becomes approximately
conservative. - Afterward, the Clausius-Clapeyron relation and
no-source assumption dictate increases in
saturation mixing ratio, qs, following a parcel
according to
21Change in parcel RH as it descend
- Define tdry as
- The relative humidity of the air parcel after a
descent time, t, is - t is the parcel age, the time since the last
moistening event - tdry is a few days and is shorter in the upper
troposphere - Assume volume of air that is saturated is small
fraction of total atmospheric volume
22Moistening Time Scale
- To define a probability density of relative
humidity, we need the probability distribution of
the time between moistening events - Proposal parcels are remoistened by
encounters that occur randomly with a fixed
probability per unit time independent of previous
history (a Poisson process) - The decay constant, tmoist, is now equal to the
mean remoistening time
23Stochastic Model RH Probability Distribution
- Assume the troposphere is very deep with constant
w, tdry and tmoist at all heights above the
interval being considered. - Combining the last 2 equations yields
- where r tdry / tmoist
- Integrating yields a cumulative distribution
24Stochastic Model RH Probability Distribution
- This equation is a bit astonishing in its
simplicity - One free parameter is needed to define the RH
distribution the ratio between tdry and tmoist. - r 1 uniform dist.,
- r lt 1 dist. is peaked at low R
- So lets assess the model with GPS RO and MLS
data
25GPS vs. Stochastic Model Cumulative RH
Distribution TROPICS
Agreement is surprisingly good! Some
disagreement at large R
- Cumulative distributions of R at three levels in
the lower and middle troposphere from GPS data
(symbols) and from the stochastic parcel model
with three values of r (lines), where r decreases
as the curves shift to the lower right.
26UARS MLS vs. Stochastic Model RH Distribution
TROPICS
- Agreement is good at 215 and 464 hPa
- Not as good at 326 hPa
- Same as previous Fig, except data is from the
UARS MLS for the upper troposphere.
27Comparison of UARS and EOS MLS Results
- UARS 215 hPa agrees better with model while EOS
316 hPa agrees better with model
28Mid-latitudes GPS UARS MLS vs. Stochastic
Model RH Histograms
- Same as previous Fig., except data from 30S-60S
and 30N-60N.
29Seasonal Cumulative Distribution (GPS-Tropics)
Jan
Jul
Apr
Oct
30Zonal Mean Humidity June 21-July 4 1995
- Derived from GPS/MET using temperatures from
ECMWF and NCEP
ltqgt
ltUgt
312D Cell Model
- The model grid contains 10 horizontal and 200
vertical locations, equally spaced in distance
and pressure respectively, with the vertical grid
ranging from 850 to 150 hPa. - Idealized, tropical clear-sky cooling profile is
also specified, equal to -1.25 K/day up to 300
hPa, then linearly decreasing with pressure to
zero at 150 hPa - Subsidence, w(p), is diagnosed to balance the
clear-sky energy budget away from convection
(Sarachik 1978, see also Folkins et al. 2002) - Net Mass detrainment
- QR is radiative heating rate, T temperature, Q
potential temperature
322D Model RH distribution vs. Observations
- Vertical Processes
- Upwelling to match subsidence
- EVAP of hygrometeors
- MIX diffusion
- Horizontal Processes
- Advection
- Diffusion
- Dissipation relaxation mixing
33Summary Conclusions
- Estimated GPS RH errors imply RH is useful for
temperatures warmer than 245K up to 9km in the
Tropics. - Deconvolution can remove negative humidities and
improve the estimate of RH PDF (but challenging) - Deconvolution can improve our estimates of the
Direct Method humidity errors which constrain a
combination of analysis temperature and GPS
refractivity errors - GPS and MLS moisture estimates are quite
complementary in their vertical coverage - Relative lack of high RH in GPS results may
indicate high RH regions in middle troposphere
are small in horizontal extent
34Conclusions
- Remarkably simple, one free parameter stochastic
model apparently explains observations - Model predicts broad distribution of RH even as r
changes - Simple 2D model captures much of the behavior but
seems to be missing a low altitude source and is
too moist at high altitudes - Data shows and 2D model predicts minimum in RH
near 400 mb - where tDry is relatively small in middle
troposphere up to 300 mb - due to decreasing dry static stability, and
- drying per unit warming decreases strongly with
decreasing temperature due to the
ClausiusClapeyron equation
35Conclusions (contd)
- Radiation controls the relative humidity
distribution - Model assumes area taken up by convection is
effectively 0 and therefore may not be important - Need to understand the physics of tmoist to
predict future climatic evolution - RH dist. may change in the future if r changes.
- Changes in cloud cover that reduce atmospheric
cooling will elevate relative humidity (slower
descent gt increase tdry gt increase r ). - Changes in organization that allow convective
moisture to rapidly spread to all parts of the
global atmosphere will reduce tmoist and increase
relative humidity, - Changes that further isolate convective systems
from other parts of the atmosphere will increase
tmoist and decrease relative humidity.