using gps ro to determine the probability density of free tropospheric relative humidity

August 23, 2005 NOAA/UCAR GPS Symposium Kursinski et al. 1

Using GPS RO to determine the Using GPS RO to determine the probability density of free probability density of free

tropospheric relative humiditytropospheric relative humidityand constrain how it is controlledand constrain how it is controlled

E. R. Kursinski1, S. Sherwood2, W. Read3

1University of Arizona, 2Yale, 3JPL

GPS Conference August 2005


OutlineOutlineMotivationAccuracy of relative humidity from GPS ROImproving the RH estimates via deconvolution

of errorsEvaluation of simple RH distribution model

– Stochastic model explanation– GPS & MLS comparisons with model– Single cell model

Summary and conclusions


Motivation for Moisture ObservationsMotivation 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


Motivation: Evaluation of a Simple ModelMotivation: 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


Testing a simple relative humidity model Testing a simple relative humidity model with observationswith 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


GPS - ECMWF Specific Humidity GPS - ECMWF Specific Humidity Profile ComparisonsProfile Comparisons

GPS (solid), ECMWF (dotted), saturation (dashed) from July 2001 (g/kg)


Subtropical Moisture Subtropical Moisture EstimatesEstimates

GPS & ECMWF comparison July 1995 10oS-25oS

Mode

Mean

OLR


Deriving Relative Humidity from GPS RODeriving 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

=> We use the Direct Method


Direct Method: Solving for water vapor given Direct Method: Solving for water vapor given NN & & TT

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

where:z height,g gravitation acceleration,m mean molecular mass of moist airT temperatureR universal gas constant

(2)

Solve for P by combining the hydrostatic and ideal gas laws and assuming temperature varies linearly across each height interval, i

i

ii

TR

gm

i

iii T

TzPzP

11

N n 1106 a1

PT a2

Pw

T2 (1)


Solving for water vapor given Solving for water vapor given NN & & TT

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.


Solving for Water Vapor given Solving for Water Vapor given NN & & TT

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.


Estimating GPS water vapor errorEstimating 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

2/123232

2

102103 xqxN

N

2/1

2

22

2

22

2

22 )2(

PB

TTR

LUB

NUB P

sT

vs

NsU


Estimated GPS Relative Humidity Error Estimated GPS Relative Humidity Error ((TropicsTropics))

Resulting GPS U error %


Negative Negative U U andand Error DeconvolutionError Deconvolution

Problem: The Direct Method U estimate can be negative => 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 + U

– Measured histogram (PDF) is convolution of the true PDF and the error PDF

– Such that PDFUmeas = PDFUtrue PDF

IF we understand the error PDF we can in theory deconvolve it from the measured PDF to recover the true PDF


U U error deconvolution (work in progress)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 PDF is symmetrical

Include 2 additional constraints:– Total Probability is conserved– Mean of distribution is conserved (=> error has zero mean)– Add these as entries in Y and rows in A

Least squares problem because # of entries in Y > X


Error Deconvolution (cont’d)Error Deconvolution (cont’d)

Finding the best PDF is a trial and error approach based upon which PDF best matches the observed

Found that a PDF that is a linear combination of gaussian and exponential works well

Summary:– Choose trial PDF

(using lin. combination of gaussian & exponential)– Calculate least squares Xls solution

– Smooth Xls if necessary

– Forward calc Y’=A 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


Deconvolution exampleDeconvolution 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.


Data Sets for Moisture Variability StudyData 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)


Now on to the Model…Now on to the Model…


Stochastic Model SummaryStochastic 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


Change in parcel RH as it descendChange in parcel RH as it descend

Define dry 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

dry 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

v

vDry L

TR

2


Moistening Time ScaleMoistening 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, moist, is now equal to the mean remoistening time


Stochastic Model RH Probability DistributionStochastic Model RH Probability Distribution

Assume the troposphere is very deep with constant , dry and moist at all heights above the interval being considered.

Combining the last equations yields

where r = dry / moist

Integrating yields a cumulative distribution


Stochastic Model RH Probability DistributionStochastic 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 dry and moist.

– r = 1 uniform dist., – r < 1: dist. is peaked at low R

So let’s assess the model with GPS RO and MLS data…


GPS vs. GPS vs. Stochastic Model Stochastic Model Cumulative RH Cumulative RH

Distribution Distribution

TROPICSTROPICS

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.

Agreement is surprisingly good!

Some disagreement at large R


UARS MLS vs. UARS MLS vs. Stochastic Model Stochastic Model RH Distribution RH Distribution

TROPICSTROPICS

Same as previous Fig, except data is from the UARS MLS for the upper troposphere.

Agreement is good at 215 and 464 hPa

Not as good at 326 hPa


Comparison of UARS and EOS MLS ResultsComparison of UARS and EOS MLS Results

UARS 215 hPa agrees better with model while EOS 316 hPa agrees better with model


Mid-latitudesMid-latitudes GPS & UARS MLS vs. Stochastic Model RH Histograms GPS & UARS MLS vs. Stochastic Model RH Histograms

Same as previous Fig., except data from 30S-60S and 30N-60N.


Seasonal Cumulative Distribution (GPS-Tropics)Seasonal Cumulative Distribution (GPS-Tropics)

Jan

Apr

Jul

Oct


Zonal Mean Humidity June 21-July 4 1995Zonal Mean Humidity June 21-July 4 1995

Derived from GPS/MET using temperatures from ECMWF and NCEP

<q> <U>


2D Cell Model2D 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, (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, potential temperature


2D Model RH distribution vs. Observations2D Model RH distribution vs. Observations

Vertical Processes

Upwelling to match subsidence

EVAP of hygrometeors

MIX: diffusion

Horizontal Processes

Advection

Diffusion

Dissipation: relaxation mixing


Summary & ConclusionsSummary & 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


Conclusions Conclusions 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 Dry 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


Conclusions (cont’d)Conclusions (cont’d) 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 moist 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 => increase dry => increase r ).

– Changes in organization that allow convective moisture to rapidly spread to all parts of the global atmosphere will reduce moist and increase relative humidity,

– Changes that further isolate convective systems from other parts of the atmosphere will increase moist and decrease relative humidity.

using gps ro to determine the probability density of free tropospheric relative humidity

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