NASA/TM–2014-104606 / Vol. 37

March 2015

Lawrence L. Takacs, Max Suarez, and Ricardo Todling

National Aeronautics and 

Space Administration 

Goddard Space Flight Center

Greenbelt, Maryland 20771

Technical Report Series on Global Modeling and Data Assimilation, 

Volume 3

Randal D. Koster, Editor 

Maintaining Atmospheric Mass and Water Balance Within 

Reanalysis
Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

1$6$VFLHQWL¿FDQGWHFKQLFDOLQIRUPDWLRQ67,SURJUDPSOD\VDNH\SDUWLQKHOSLQJ1$6$PDLQWDLQWKLV

LPSRUWDQWUROH

7KH1$6$67,SURJUDPRSHUDWHVXQGHUWKHDXVSLFHV

RIWKH$JHQF\&KLHI,QIRUPDWLRQ2I¿FHU,WFROOHFWV

organizes, provides for archiving, and disseminates 

1$6$¶V67,7KH1$6$67,SURJUDPSURYLGHVDFFHVV

to the NASA Aeronautics and Space Database and its 

SXEOLFLQWHUIDFHWKH1$6$7HFKQLFDO5HSRUW6HUYHU

WKXVSURYLGLQJRQHRIWKHODUJHVWFROOHFWLRQVRIDHURQDXWLFDODQGVSDFHVFLHQFH67,LQWKHZRUOG5HVXOWV

DUHSXEOLVKHGLQERWKQRQ1$6$FKDQQHOVDQGE\

1$6$LQWKH1$6$67,5HSRUW6HULHVZKLFKLQFOXGHV

WKHIROORZLQJUHSRUWW\SHV

• 7(&+1,&$/38%/,&$7,215HSRUWVRI

FRPSOHWHGUHVHDUFKRUDPDMRUVLJQL¿FDQWSKDVHRI

UHVHDUFKWKDWSUHVHQWWKHUHVXOWVRI1$6$3URJUDPV

DQGLQFOXGHH[WHQVLYHGDWDRUWKHRUHWLFDODQDO\VLV

,QFOXGHVFRPSLODWLRQVRIVLJQL¿FDQWVFLHQWL¿FDQG

WHFKQLFDOGDWDDQGLQIRUPDWLRQGHHPHGWREHRI

FRQWLQXLQJUHIHUHQFHYDOXH1$6$FRXQWHUSDUWRI

SHHUUHYLHZHGIRUPDOSURIHVVLRQDOSDSHUVEXWKDV

OHVVVWULQJHQWOLPLWDWLRQVRQPDQXVFULSWOHQJWKDQG

H[WHQWRIJUDSKLFSUHVHQWDWLRQV

• 7(&+1,&$/0(025$1'806FLHQWL¿F

DQGWHFKQLFDO¿QGLQJVWKDWDUHSUHOLPLQDU\RURI

VSHFLDOL]HGLQWHUHVWHJTXLFNUHOHDVHUHSRUWV

ZRUNLQJSDSHUVDQGELEOLRJUDSKLHVWKDWFRQWDLQ

PLQLPDODQQRWDWLRQ'RHVQRWFRQWDLQH[WHQVLYH

DQDO\VLV

• &2175$&7255(32576FLHQWL¿FDQGWHFKQLFDO

¿QGLQJVE\1$6$VSRQVRUHGFRQWUDFWRUVDQG

grantees.

• &21)(5(1&(38%/,&$7,21&ROOHFWHG

SDSHUVIURPVFLHQWL¿FDQGWHFKQLFDOFRQIHUHQFHV

V\PSRVLDVHPLQDUVRURWKHUPHHWLQJVVSRQVRUHGRU

FRVSRQVRUHGE\1$6$

• 63(&,$/38%/,&$7,216FLHQWL¿FWHFKQLFDO

RUKLVWRULFDOLQIRUPDWLRQIURP1$6$SURJUDPV

SURMHFWVDQGPLVVLRQVRIWHQFRQFHUQHGZLWK

VXEMHFWVKDYLQJVXEVWDQWLDOSXEOLFLQWHUHVW

• 7(&+1,&$/75$16/$7,21(QJOLVKODQJXDJH

WUDQVODWLRQVRIIRUHLJQVFLHQWL¿FDQGWHFKQLFDO

PDWHULDOSHUWLQHQWWR1$6$¶VPLVVLRQ

6SHFLDOL]HGVHUYLFHVDOVRLQFOXGHRUJDQL]LQJDQG

SXEOLVKLQJUHVHDUFKUHVXOWVGLVWULEXWLQJVSHFLDOL]HG

UHVHDUFKDQQRXQFHPHQWVDQGIHHGVSURYLGLQJKHOS

GHVNDQGSHUVRQDOVHDUFKVXSSRUWDQGHQDEOLQJGDWD

H[FKDQJHVHUYLFHV)RUPRUHLQIRUPDWLRQDERXWWKH

1$6$67,SURJUDPVHHWKHIROORZLQJ

• $FFHVVWKH1$6$67,SURJUDPKRPHSDJHDW

KWWSZZZVWLQDVDJRY

• (PDLO\RXUTXHVWLRQYLDWKH,QWHUQHWWR

KHOS#VWLQDVDJRY

• )D[\RXUTXHVWLRQWRWKH1$6$67,+HOS'HVNDW

443-757-5803

• 3KRQHWKH1$6$67,+HOS'HVNDW

• :ULWHWR

1$6$67,+HOS'HVN

1$6$&HQWHUIRU$HUR6SDFH,QIRUPDWLRQ

7115 Standard Drive 

+DQRYHU0'

1$6$67,3URJUDPLQ3URILOH
National Aeronautics and 

Space Administration 

Goddard Space Flight Center

Greenbelt, Maryland 20771

NASA/TM–2014-104606 / Vol. 37

March 2015

Lawrence L. Takacs 

NASA’s Goddard Space Flight Center, Greenbelt, Maryland

Max Suarez 

NASA’s Goddard Space Flight Center, Greenbelt, Maryland 

Ricardo Todling 

NASA’s Goddard Space Flight Center, Greenbelt, Maryland

Technical Report Series on Global Modeling and Data Assimilation, 

Volume 37

Randal D. Koster, Editor

Maintaining Atmospheric Mass and Water Balance Within 

Reanalysis
$YDLODEOHIURP

1$6$&HQWHUIRU$HUR6SDFH,QIRUPDWLRQ

7115 Standard Drive 

+DQRYHU0'

1DWLRQDO7HFKQLFDO,QIRUPDWLRQ6HUYLFH

3RUW5R\DO5RDG

6SULQJ¿HOG9$3ULFH&RGH$

Level of Review: This material has been technically reviewed by technical management

7UDGHQDPHVDQGWUDGHPDUNVDUHXVHGLQWKLVUHSRUWIRULGHQWLILFDWLRQRQO\7KHLUXVDJHGRHVQRW

FRQVWLWXWHDQRIILFLDOHQGRUVHPHQWHLWKHUH[SUHVVHGRULPSOLHGE\WKH1DWLRQDO$HURQDXWLFVDQG

Space Administration.

Notice for Copyrighted Information

7KLVPDQXVFULSWLVDZRUNRIWKH8QLWHG6WDWHV*RYHUQPHQWDXWKRUHGDVSDUWRIWKHRIILFLDOGXWLHV

RIHPSOR\HHVRIWKH1DWLRQDO$HURQDXWLFVDQG6SDFH$GPLQLVWUDWLRQ1RFRS\ULJKWLVFODLPHG

LQWKH8QLWHG6WDWHVXQGHU7LWOH86&RGH$OORWKHUULJKWVDUHUHVHUYHGE\WKH8QLWHG6WDWHV

*RYHUQPHQW$Q\SXEOLVKHUDFFHSWLQJWKLVPDQXVFULSWIRUSXEOLFDWLRQDFNQRZOHGJHVWKDWWKH8QLWHG

6WDWHV*RYHUQPHQWUHWDLQVDQRQH[FOXVLYHLUUHYRFDEOHZRUOGZLGHOLFHQVHWRSUHSDUHGHULYDWLYH

ZRUNVSXEOLVKRUUHSURGXFHWKHSXEOLVKHGIURPRIWKLVPDQXVFULSWRUDOORZRWKHUVWRGRVRIRU

8QLWHG6WDWHV*RYHUQPHQWSXUSRVHV.
Abstract 

This report describes the modifications implemented into the Goddard Earth Observing System 

Version-5 (GEOS-5) Atmospheric Data Assimilation System (ADAS) to maintain global 

conservation of dry atmospheric mass as well as to preserve the model balance of globally 

integrated precipitation and surface evaporation during reanalysis. Section 1 begins with a 

review of these global quantities from four current reanalysis efforts. Section 2 introduces the 

modifications necessary to preserve these constraints within the atmospheric general 

circulation model (AGCM), the Gridpoint Statistical Interpolation (GSI) analysis procedure, and 

the Incremental Analysis Update (IAU) algorithm. Section 3 presents experiments quantifying 

the impact of the new procedure. Section 4 shows preliminary results from its use within the 

GMAO MERRA-2 Reanalysis project. Section 5 concludes with a summary. 

2 
1) Introduction 

A key goal of recent atmospheric reanalysis efforts has been to extract statistically significant 

signals from within Data Assimilation System (DAS) products that are relevant to climate 

change. By using the most scientifically advanced but “frozen” systems during reanalysis, we 

expect to remove the impact of improved model physics and/or dynamics over time and 

hopefully more clearly determine trends in the actual climate based on the assimilated 

observational record. For example, trends in global surface temperature should be detected by 

trends in globally integrated water content and the associated estimates of precipitation and 

surface evaporation. Complicating this scenario, however, are changes in observational 

instrument and satellite technology over the reanalysis period and the model response 

associated with those changes. As noted in Bosilovich and Robertson (2011), the latest 

reanalyses are still significantly impacted by changes in the observing system with regard to 

globally integrated precipitation estimates. Figure 1.1 shows the time series of the globally 

integrated evaporation (E) and precipitation (P) from four existing reanalyses: 1) MERRA 

(Rienecker et al., 2011), 2) ERA-Interim (Dee et al., 2011), 3) JRA-55 (Ebita et al., 2011), and 4) 

NCEP-CFSR (Saha et al., 2010). We clearly see imbalances between P and E from all four cases 

that may or may not be physically linked to changes in total atmospheric water content. 

Imbalances are of both signs and can even change sign within a single reanalysis. Changes in 

the globally integrated precipitation tend, in general, to be more volatile than globally 

integrated evaporation, and they may be more susceptible to changes in atmospheric water 

vapor content due to changes in the observational satellite database over time. 

In addition to global estimates of precipitation, there is also considerable interest in the 

estimate of total atmospheric mass (e.g., Trenberth and Smith, 2005; Hoinka, 1998; Trenberth 

and Guillemot, 1994; Trenberth, 1991; Trenberth et al., 1987). As suggested by Trenberth and 

Smith (2005), the total mass of the atmosphere can and should be utilized as a constraint in 

global analyses in addition to being of interest in its own right. As they and the other studies 

have pointed out, the globally integrated dry atmospheric mass to first order should be 

conserved (ignoring such effects as burning of fossil fuels, anthropogenic sources and sinks, the 

outgassing in volcanoes, and loss to space). In each of the four reanalyses in Fig. 1.1, however, 

there is no fundamental mechanism to ensure that resulting analyzed states maintain 

conservation of globally integrated dry atmospheric mass. Conservation of dry mass implies 

that there is a consistent relationship between changes in total mass and changes in total water 

content. 

3 
Figure 1.1: Monthly mean globally integrated precipitation and surface evaporation from MERRA,

ERA-Interim, JRA-55, and NCEP-CFSR Reanalyses. 

4 
For our purposes we define total mass (in units of hydrostatic pressure) as simply the sum of 

the atmospheric dry mass plus the mass of the total water content: 

where is the total water content defined as the sum of the vapor, liquid, and frozen 

components: 

, 

and is the specific mass of constituent k. Figure 1.2 shows the time series from the four 

reanalyses of the climatological seasonal cycle of total water content and their corresponding 

monthly-mean time tendency, in addition to the monthly mean total water content anomalies. 

The climatological seasonal cycles of total water content are all well analyzed, with small 

variations in magnitude of the annual mean values between the reanalyses. The climatological 

time tendencies of total water content, however, are virtually identical. With regard to the 

monthly mean anomalies, there is a high degree of correlation between MERRA and NCEPCFSR, while ERA-Interim and JRA-55 are more closely related. The MERRA and NCEP-CSFR 

anomalies show a distinct rise in total water content over the reanalysis period, which is 

consistent with the general rise in the prescribed temperatures used as boundary conditions 

during the reanalysis. 

To delineate more clearly the imbalance between E and P, Figure 1.3 compares the implied

model-generated globally-integrated water tendency (i.e., E-minus-P, or E-P) on the same scale 

as the actual tendency of globally integrated assimilated water vapor. We see that the implied 

water vapor tendency from the model derived E-P residual is much larger than the actual water 

vapor tendency. Assuming that the models used in the reanalyses contain no spurious sources 

or sinks of water content, the imbalance between the implied water vapor tendency (E-P) and 

the actual water vapor tendency must be due to water added or removed by the analysis. 

5 
As noted earlier, to conserve globally integrated dry mass there must be a physically consistent 

relationship between changes in total mass and changes in total water content. We have seen, 

however, that the four reanalyses violate the consistency of implied total water content 

tendency (through E-P) compared with actual assimilated total water tendency. This gives rise 

to questions regarding the system’s total mass and its separation into dry and wet components. 

Figure 1.4 shows the time series of the climatological seasonal cycle of total mass (in terms of 

surface pressure) as well as the monthly mean anomalies from the four reanalyses. In addition 

we show the time series of the dry and wet components separately. The climatological 

seasonal cycle of total mass clearly reflects the seasonal cycle of total water content. However, 

the magnitudes of the annual mean values vary substantially between the reanalyses. Most 

disturbing is the lack of conservation of dry mass and the high correlation between the total 

Figure 1.2: Climatologies of total water content (top left) and tendency (top right), plus the total 

water content anomaly (bottom) from MERRA, ERA-Interim, JRA-55, and NCEP-CFSR Reanalyses. 

6 
and dry mass anomalies rather than between the total and wet anomalies. The lack of dry mass 

conservation and the imbalance between globally integrated P and E from all four reanalyses 

are physically unrealistic; these problems serve as motivation to improve the GMAO 

Atmospheric Data Assimilation System (ADAS) for the upcoming GMAO MERRA-2 reanalysis. 

2) Modifications to Model and Analysis 

2a) Adjustment to the model’s mass balance: 

As shown in Section (1), the reanalysis products from all centers exhibit a fairly consistent 

representation of the seasonal cycle of total mass due to the seasonal cycle of total water 

content. In the MERRA system this was driven completely by the analyzed surface pressure 

since the GEOS-5 AGCM had no pressure change due to changes in water content from moist 

physics. Further, since conservation of total mass is guaranteed in the dynamical core, all time 

fluctuations of total mass are due to the analysis increment of surface pressure. Therefore, as a 

first step toward the development of physically-based dry mass conservation, the AGCM had to 

be modified to incorporate changes in total pressure due to changes in water content from the 

moist physics. 

In the GEOS-5 AGCM, moist physics parameterizations consist of time tendencies of specific 

humidity (in the form of vapor: , liquid: , and ice: ) due to contributions from 

convection and large-scale rain, from turbulence, and from stratospheric chemistry. In the 

model used for MERRA, these parameterizations preserve the mass of each layer, . This 

behavior is retained in the new model but a correction is made to the layer masses to properly 

account for the mass change due to evaporation and precipitation: 

 = , 

where is the specific mass of total water produced by the physics, and the superscripts b

and a indicate quantities before and after the mass corrections are applied. Finally, a correction 

is made to the specific masses of all constituents, including the water specific masses, to 

account for total mass changes associated with any change in total water produced by the 

physics. This correction is done on a layer by layer basis to conserve constituent mass in each 

layer. If is the specific mass of constituent k upon exiting the physics, it is multiplied by the 

ratio of the layer masses before and after the change in total water is accounted for:

7 
where is the layer mass of constituent k as the physics left it and is identical to , 

the constituent mass after the change in total mass due to water changes and the correction to 

.

With this modification to the moist physics, the globally integrated time-tendency of the total 

mass of the atmosphere is governed by the time-tendency of the moist component, thereby 

leaving the dry mass globally conserved. Since the model now retains whatever initial value it 

sees, a value must be chosen for the globally integrated dry mass (or surface pressure 

equivalent) in the model’s initial conditions. Trenberth and Smith (2005) documented global 

dry mass monthly values from three different reanalyses (ERA-40, ERA-15, and NCEP-NCAR) 

over the period from 1950-2000 to range from 982 to 984 hPa. Figure 2.1 shows the dry 

surface pressures from the four reanalyses discussed in Section 1. Based on these more recent 

reanalyses, a value of 983.24 hPa was chosen for initialization of GMAO model initial conditions 

and the MERRA-2 system. 

Using the developments outlined above, Figure 2.2 shows the DJF and JJA seasonal 

climatologies of the surface pressure tendency due to moist physics from a 30-year 1⁄2-deg 

GEOS-5 AGCM AMIP simulation. We clearly see the decrease of mass associated with 

precipitation in regions of tropical convection and mid-latitude storm tracks, as well as the 

increase of mass in regions of surface evaporation and subsidence. 

8 
Figure 1.3: Implied (E-P) vs actual monthly mean globally integrated water tendency from MERRA, 

ERA-Interim, JRA-55, and NCEP-CFSR Reanalyses. 

9 
Figure 1.4: Climatology and monthly mean anomalies of globally integrated mass from MERRA, ERAInterim, JRA-55, and NCEP-CFSR Reanalyses. Also shown is the partition of dry and moist 

components. 

10 
Figure 2.2: Seasonal climatologies of the surface pressure tendency due to moist physics. 

Figure 2.1: Globally integrated dry surface pressure from four reanalyses as well as the value chosen for 

the MERRA-2 system. 

11 
Figure 2.3 shows the time-series of globally integrated mass within the 30-year 1⁄2-deg GEOS-5 

AGCM AMIP simulation. While the model used for MERRA had no seasonal cycle of total 

pressure (not shown), Figure 2.3 shows a very realistic climatological seasonal cycle of total 

pressure which is perfectly correlated with the climatological seasonal cycle of total water 

content. The 30-year time-series of globally integrated surface pressure anomalies shows that 

dry mass is perfectly conserved during the run. All trends within the total pressure are solely 

due to trends within the total water content (total and wet components overlay exactly). It is 

interesting to note that the trend in total water content is highly correlated (as shown) with the 

trend in the prescribed 1-deg SST (Reynolds et al., 2002) boundary condition. 

Figure 2.3: Climatology and monthly mean anomalies of globally integrated mass from the GEOS-5 

AGCM AMIP simulation. Also shown is the partition of dry and moist components. (Note: SST 

anomalies (K) are scaled on the right) 

12 
Figure 2.4 shows the time-series of globally integrated total precipitation and surface 

evaporation from the same 30-year GEOS-5 AGCM climate simulation. In this pure model run 

we see the consistent long-term balance between sources and sinks of total water content. 

These results show that the updated model includes the necessary improvements needed for 

dry mass conservation and can maintain proper balance between globally integrated 

precipitation and evaporation. 

2b) Additional Analysis Constraint: 

If, as discussed earlier, the data assimilation system is expected to conserve total dry mass, not 

only do we need the AGCM to preserve this quantity, the atmospheric analysis must also be 

constrained to preserve dry mass. Unlike the version of GSI used for MERRA, the present 

version of GSI (Kleist et al., 2009) includes a trigger for an additional constraint to penalize 

changes in dry mass (J. C. Derber and D. T. Kleist, pers. comm.). Originally, this feature was 

implemented to work within the context of the minimization option that uses the preconditioned conjugate gradient (PCG) of Derber and Rosati (1989). However, presently the 

GEOS-5 ADAS uses the bi-conjugate gradient (BiCG) algorithm of El Akkraoui et al. (2013), which 

is available as one of the alternative minimization procedures in GSI. The main difference 

between the PCG and BiCG implementations is in the calculation of the step defining the 

steepest descent at each iteration of the minimization, with the BiCG using a simple gradient 

norm ratio which works well when the inner loop of three-dimensional, variational (3D-Var) 

analysis is linearized, as in GEOS-5 ADAS. The analysis software has then been revised to allow 

for the dry mass constraint to apply regardless of the minimization procedure employed by GSI. 

Figure 2.4: Monthly mean globally integrated precipitation and surface evaporation from the GEOS5 AGCM AMIP simulation. 

13 
The implementation of this constraint amounts to introducing an additional term, Jd, to the 

usual analysis cost function: 

 J = Jb + Jo + Jd , 

where Jb and Jo represent the typical background and observation cost terms of a first-guess-atappropriate-time 3D-Var analysis (e.g., see Massart et al., 2010). Explicitly, the term ensuring 

minimal changes to total dry mass is written as 

 Jd = J ( GqGqw )2

where Gq = 6k Gqk GAGpk is the increment to total mass calculated over all vertical levels of the 

background; Gqw = 6k [Gqk

v

 + Gqk

l + Gqki

] GA Gpk is the incremental contribution of vapor, liquid 

and ice water contents to total mass, with GA and Gpk representing the horizontal and vertical 

grid weights, respectively; and the parameter J is used to control the strength of the penalty. 

Since this term is applied as a weak-constraint to the minimization problem, dry-mass 

conservation is only approximately guaranteed. 

Figure 2.5: Dry-mass and wet-mass time series for two experiments: (i) when a cycled analysis is 

constrained to conserve dry mass (red curves), and (ii) when a the analysis is performed without 

constraint from the background of (i) (black curves). Dry surface pressure is shown by the open 

circles. Also shown are the analysis increments on dry mass (closed-red and black circles), and 

analysis increments on mass water content (open squares). Surface pressure scale is shown on the 

left vertical axis; increment scale is shown on the right vertical axis. 

14 
Experimentation suggests that J= 5x104 is enough to keep dry mass nearly conserved. As an 

illustration, Fig. 2.5 shows time series of analysis increments for two cases: (i) a cycling 

experiment in which the analysis includes the constraint on dry mass; and (ii) a set of noncycling experiments in which the analysis does not include the constraint. In both cases the GSI 

analyses see exactly the same background states, and in both cases, the GEOS-5 AGCM 

producing the background fields conserves dry mass following the modifications of Section 2a. 

The figure shows analyzed dry mass as the open-circle lines; we see that without the constraint 

(black) the analysis does not maintain a constant dry mass, whereas it does when the constraint 

is applied (red), keeping it at the set value of 983.24 hPa. The dry and wet contributions to the 

mass increment are shown by the closed-circle-dashed and open-square-dotted curves, 

respectively. Over the short 9-day period considered here, we see that without the dry-mass 

constraint the analyzed dry-mass changes as the water content increment changes (black). The 

more physically desirable behavior is shown by the red curves, with the dry mass left 

unchanged and only the wet contribution varying. 

Also pertinent to the present discussion is the fact that the moisture control variable used in 

the most recent version of GSI (and in the upcoming MERRA-2) is different from that used in its 

MERRA version. The latter used the so-called pseudo-relative humidity (RH; Dee and da Silva, 

2003) whereas the former uses normalized RH (Holm 2003). As it turns out, independent 

examination finds that pseudo-RH makes humidity increments more sensitive to use of the 

AMSU-A window channels (pers. comm. William McCarty and Andrea Mold). These channels 

are used in both MERRA and NCEP-CFSR, with both using the same pseudo-RH humidity control 

variable choice. This sensitivity is noticeable in the E-P time series shown in Fig. 1.3, and the 

jump around the end of 1998 corresponds to when the first AMSU-A instrument, on NOAA-15, 

became available (see table 1 in Rienecker et al., 2011). Indeed, neither ERA-Interim nor JRA-55 

shows the jump and sensitivity – in the case of ERA-Interim, not only is the control variable 

normalized RH, but also the near-surface and window channels, 1-4 and 15, of AMSU-A are not 

used (Dee et al., 2011). 

2c) Adjustment to the Incremental Analysis Update:

In a traditional implementation of an intermittent assimilation cycle, the analysis provides a 

correction to the model initial condition. In this case, with the model conserving dry mass and 

with the analysis constrained to allow only changes in the water content that are consistent 

with the change in total mass, the two features discussed in Sections 2a and 2b keep dry mass 

nearly unchanged. However, the GEOS-5 ADAS uses the incremental analysis update (IAU) 

method of Bloom, et al. (1996). Instead of correcting the initial condition, IAU turns the analysis 

increment into a tendency term that is applied to the model during the assimilation cycle. In 

15 
this context, the assimilation in GEOS-5 quickly fails to conserve dry mass given the constancy 

of the surface pressure and moisture increments over the 6-hour assimilation window. In 

addition, as shown in Section 1, analysis increments of globally integrated mass and moisture 

that fail to vanish may lead to imbalances between globally integrated evaporation and 

precipitation. As a result, globally integrated precipitation amounts become very susceptible to 

changes in the observational database used by the analysis. Therefore, the IAU procedure has 

been modified such that the globally integrated analysis increments of mass and moisture 

vanish, thereby constraining globally integrated dry mass to be conserved. It should be noted 

that this modification forces all trends in global mass and water content to be driven purely by 

the model physics, and are only indirectly influenced by the analyzed states of temperature, 

wind, moisture, and pressure. 

Following the procedure used in IAU, we begin with the addition of a constant analysis 

increment of surface pressure and moisture at every timestep within the 6-hour analysis cycle. 

At each time-step we compute the smallest departure from the prescribed increment (in a 

least-squares sense) that will produce vanishing global integrals of the increments. Defining 

the two functionals, and , which measure the globally integrated squared departure from 

initial values plus the strong constraints of vanishing increments, we have: 

Here and represent the surface pressure and moisture of the background at time , 

and represent the surface pressure and moisture that would be produced by adding the 

original GSI analysis increments and , and and represent the target values 

satisfying the constraints. Scaling variables and are introduced for generality. Solving the 

above system through minimization of and yields: 

16 
While various options were evaluated, choosing scaling variables defined by:

yields the simple solutions: 

Here represents global horizontal integrals. These updated and constrained values of 

pressure and moisture will now be used in a series of GEOS-5 ADAS experiments to examine 

their impact on dry mass conservation and P & E balance. 

3) Experimental Design 

3a) ADAS Evaluation 

The model and analysis updates described in Section 2 have been incorporated for use in a 

series of GEOS-5 ADAS experiments. As a baseline for these tests, data were taken from an 

interim reanalysis prior to MERRA-2 called GMAO RPIT (Lucchesi, R., 2013), which ran from 

January 1980 through December 2011. As shown in Fig. 3.1, the GMAO RPIT reanalysis exhibits

dramatic changes in globally integrated precipitation due to changes in the satellite observing 

systems during this time. The nature of jumps such as the one seen in 2008 is not the same as 

the jump seen in MERRA (top panel of Fig. 1.3); here the window channels of AMSU-A are not 

used, and GSI uses normalized RH for its humidity control variable. Though not yet fully 

diagnosed, the jump in precipitation in Fig. 3.1 is most likely associated with the Infrared 

Atmospheric Sounding Interferometer (IASI) on the METOP-A satellite, which was introduced in 

the RPIT experiment at the beginning of August 2008. Whatever the reason, the fact remains 

that the default settings in the GEOS-5 ADAS used for RPIT make the water behavior too 

17 
sensitive to changes in the observing system. The RPIT experiment provides a good case study 

for our purposes here, and we therefore begin this section with a more detailed examination of 

the RPIT reanalysis for two periods in particular: March 2003 and March 2009. 

Figure 3.2 depicts the globally integrated surface pressure (together with its wet and dry 

components) based on hourly data from the RPIT reanalysis for the months of March 2003 and 

March 2009. Two issues are readily apparent: 1) a lack of total correlation between the total 

and wet surface pressures, and 2) a close correlation between the total and dry surface 

pressure anomalies. The lack of correlation (issue 1) is related to inconsistencies between the 

analysis increment of surface pressure and the vertically integrated analysis increment of 

moisture (see Figure 3.3) such that their globally integrated values do not match: 

IAU Tendency March 2003 March 2009 

d(PS)/dt 0.000908 -0.000306 

d(QINT)/dt 0.183114 -0.239858 

Figure 3.1: Monthly mean globally integrated precipitation and evaporation from the GMAO RPIT 

Reanalysis . 

Table 3.1: Globally integrated analysis increments of surface pressure (PS) and column water (QINT) 

in units of hPa/day. 

18 
Figure 3.4 shows the hourly time-series of the globally integrated precipitation and evaporation 

from the GMAO RPIT reanalysis for the two periods: top panel for March 2003; middle panel for 

March 2009. It is interesting to note that the mean value of surface evaporation is just under 

2.9 mm/day for both periods while the mean precipitation value ranges from 3.1 mm/day in 

March 2003 to 2.6 mm/day in March 2009. Since the total vertically integrated water budget is 

physically based on the E-P difference, the implied water tendency is quite large for these two 

periods (shown in the bottom panel of 3.4 using 24-hour running means) with significant nonzero monthly means. The actual vertically integrated precipitable water tendency, however, is 

much smaller with near-zero monthly means. This difference between actual and implied 

water tendency is due to the influence of the observationally-based analysis increments and 

Figure 3.2: Globally integrated surface pressure and its wet and dry partitions from the GMAO RPIT 

Reanalysis for March 2003 and March 2009. 

19 
illustrates the degree to which analysis increments may influence precipitation amounts during 

data assimilation reanalyses. 

Figure 3.3: Monthly mean analysis increments of surface pressure and vertically integrated 

moisture (in hPa/day) from the GMAO RPIT reanalysis for the periods March 2003 and March 2009. 

20 
Figure 3.4: Globally integrated evaporation and precipitation from the GMAO RPIT Reanalysis for 

March 2003 (a) and March 2009 (b). Also shown (c) is the implied (E-P) vs actual water tendency. 

21 
Greater insight may be gained by examining the vertical structure of the zonal mean analysis 

and convective and large-scale moisture tendencies. Figure 3.5 shows the monthly mean, zonal 

mean moisture tendencies from the two time periods as well as their difference. We see that 

the structure of the analysis tendency from IAU is significantly different between the two years. 

In March 2009, the removal of moisture in the upper levels (~500 hPa) due to the analysis is 

much stronger than in March 2003. As a result, removal by model convection and large-scale 

processes during March 2009 is significantly impacted resulting in a drop in total precipitation. 

In addition, the analysis tendency in lower levels (~850 hPa) is significantly greater in March 

2003 than in March 2009. It is difficult for the model to retain the additional moisture and, as a 

result, it is rained out by convective and large-scale processes. Both of these features tend 

toward producing more precipitation during March 2003 compared to March 2009. This is 

consistent with the time-series of total precipitation shown in Fig. 3.4. 

We have now seen that by failing to constrain the GMAO RPIT reanalysis to conserve dry mass 

and by allowing non-vanishing global integrals of moisture and surface pressure increments, the 

GEOS-5 ADAS produces inconsistencies between total and wet surface pressure (due to 

spurious sources and sinks of dry mass) as well as analysis moisture increments which 

dramatically impact moist physical parameterizations within the model. Following the 

development in Section 2, we reran each period using the updated GEOS-5 ADAS (together with 

a preliminary version of the observational database targeted for the GMAO MERRA-2 

reanalysis) using two separate configurations: 

a) Experiment ADAS-1: This configuration (without global constraints) employs the 

update to the GSI analysis (thereby penalizing increments which produce deviations 

in dry mass) together with the AGCM update for pressure changes due to moist 

physics. Conservation of dry mass is not guaranteed since the IAU procedure is not 

modified. 

b) Experiment ADAS-2: This configuration (with global constraints) employs the GSI 

and model updates as well as the assimilation update, thereby ensuring 

conservation of dry mass and vanishing globally integrated pressure and moisture 

increments. 

Both of these configurations use normalized RH for the GSI humidity control variable. In each 

case, a two week spin-up is used with February 15 initial conditions from the GMAO RPIT 

reanalysis. 

22 
Figure 3.5: Zonal mean moisture tendencies (g/kg/day) from the analysis (ANA; left column) and convective 

and large-scale rain (MST; right column) from the GMAO RPIT Reanalysis for March 2003 and March 2009. 

Bottom panels show differences.

23 
Figure 3.6 depicts the globally integrated surface pressure (together with its wet and dry 

components) based on hourly data from Experiment ADAS-1 for the months of March 2003 and 

March 2009, while Fig. 3.7 shows results from Experiment ADAS-2. Within ADAS-1 we see that 

the GSI penalty on globally integrated dry mass departures has made a significant impact 

toward improving dry mass conservation and consistency between total and wet surface 

pressure tendencies. The original IAU procedure of Bloom et al. (1996) amounts to a constant 

tendency during the assimilation period and is thus not able to completely conserve total dry 

mass. 

Figure 3.6: Globally integrated surface pressure and its wet and dry partitions from Experiment 

ADAS-1 (without Constraints) for March 2003 and March 2009. 

24 
With the modified IAU strategy discussed in Section 2c, we see perfect conservation of globally 

integrated dry mass in ADAS-2 (Fig. 3.7). Note that the tendency of wet surface pressure is 

nearly identical between the ADAS-1 and ADAS-2 experiments. Only the dry mass values have 

been significantly affected. This is important since the GSI analysis is allowed to influence the 

total water content of the atmosphere in ADAS-1 (i.e., no restrictions on globally integrated 

increments) while total water content is purely model-driven in ADAS-2. This is illustrated more 

clearly when looking at the globally integrated total precipitable water vapor (Fig 3.8). 

Here we see that the difference between the two experiments (i.e., with and without global 

constraints) is smaller than the difference between either run and the original GMAO RPIT 

reanalysis. 

Figure 3.7: Globally integrated surface pressure and its wet and dry partitions from Experiment 

ADAS-2 (with Constraints) for March 2003 and March 2009. 

25 
In the early part of the 2003 period (top panel of Fig. 3.8) we see that with the more recent 

version of GEOS ADAS (used for ADAS-1 and ADAS-2), the two experiments showed a spin up 

(down) behavior for total precipitable water that differs from that seen in RPIT. This is 

attributed to a combination of model, analysis, and observing system changes that seem to 

affect the 2003 period more than 2009 (bottom panel of same figure). 

Figures 3.9 and 3.10 show the hourly time-series of the globally integrated precipitation and 

evaporation from, respectively, ADAS-1 and ADAS-2. In these experiments there is less contrast 

between the March 2003 and March 2009 periods with respect to values of precipitation. We 

still see, however, a significant difference between the implied and actual vertically integrated 

water tendency within ADAS-1, indicating that the analysis increments still contribute 

significantly to global budgets. Within ADAS-2, the implied and actual water tendencies are in 

perfect agreement due to the constraint of vanishing global integrals of vertically integrated 

moisture increments. 

26 
Figure 3.8: Globally integrated total precipitable water from GMAO RPIT as well as from Experiments 

ADAS-1 (without Constraints) and ADAS-2 (with Constraints) for March 2003 (top) and March 2009 

(bottom). 

27 
Figure 3.9: Globally integrated evaporation and precipitation from Experiment ADAS-1 for March 

2003 (a) and March 2009 (b). Also shown (c) is the implied (E-P) vs actual water tendency. 

28 
Figure 3.10: Globally integrated evaporation and precipitation from Experiment ADAS-2 for March 

2003 (a) and March 2009 (b). Also shown (c) is the implied (E-P) vs actual water tendency. 

29 
3b) REPLAY Evaluation 

From the results in Section 3a we have seen that the modified IAU procedure produces a 

conservation of globally integrated dry atmospheric mass as well as a physically-based balance 

between globally integrated evaporation and precipitation. As a result, the time-series of 

globally integrated P and E become less sensitive to changes in the observational satellite 

database used during the assimilation (cf. Figure 3.1) and, therefore, should be more reliable 

for studies of long-term trends and interannual variability. 

To examine this impact in an expedient manner we employ the “REPLAY” option within the 

GEOS-5 AGCM. The REPLAY mechanism is similar to the standard GEOS-5 ADAS procedure in 

the sense that it uses IAU to force the assimilation. REPLAY is different from the standard 

GEOS-5 ADAS in that it uses pre-existing analyses to produce IAU increments. The assimilation 

produced from using REPLAY is thus a blend of the given analysis combined with the 

atmospheric model in use. This mechanism has been shown to reproduce the original analysis 

quite well. A singular advantage of REPLAY is that the model used may be different from that 

originally employed to generate the analysis. In both the GEOS-5 ADAS and the GEOS-5 AGCM 

with REPLAY, analysis increments are produced and used to force the model during the socalled IAU Corrector step (see Rienecker, et al., 2011). REPLAY may thus be used by modelers 

as a first step toward correcting systematic biases revealed through an examination of the IAU 

increments and toward understanding the possible impact of model changes in the data 

assimilation system. 

Using this mechanism, we REPLAY the GMAO RPIT reanalysis from January 2000 through 

December 2011 using the updated GEOS-5 AGCM (targeted for MERRA-2) under configurations 

similar to those described in Section 3a (i.e., with and without the global constraints imposed 

on the IAU increments of and ). For the REPLAY without global constraints we also removed 

the model update providing pressure changes from moist physics (thereby mimicking the RPIT 

configuration more precisely). 

Figure 3.11 shows the climatology and monthly mean anomalies of the globally integrated mass 

and its partition into dry and wet components from the REPLAY without global constraints.

Running in this configuration is comparable to the procedure used in the original GMAO RPIT 

reanalysis that contained no mechanism for dry mass conservation. Results from the REPLAY 

run are essentially identical to those from the original GMAO RPIT analysis (not shown). We

again observe (top panel) a lack of total correlation between the climatological seasonal of total 

(black curve) and wet (cyan curve) mass, as well as a spurious dry mass anomaly (bottom 

30 
panel, orange 

curve).

Figure 3.12 shows the same result but for the REPLAY with the global constraints invoked. Here 

we see that dry mass is perfectly conserved during the run, giving rise to a perfect correlation 

between the total and wet climatological seasonal cycles (top) and anomalies (bottom). 

Because the REPLAY assimilation is not allowed to influence the fixed analysis, we see that the 

trend in the wet component very nearly matches the trend in the REPLAY without constraints. 

Figure 3.11: Climatology and monthly mean anomalies of globally integrated mass from the GEOS5-

AGCM REPLAY of the RPIT Reanalyses. This REPLAY did not use global constraints. 

31 
Figure 3.13 repeats the plot for the globally integrated precipitation and evaporation from the 

GMAO RPIT reanalysis in addition to showing results from the two REPLAY experiments. We 

see that REPLAY without global constraints produces the same signature with respect to the P 

and E balance as does RPIT. There are small differences in precipitation amounts compared 

Figure 3.12: Climatology (top: black curve for total; cyan for wet) and monthly mean anomalies 

(bottom: black for total; cyan for wet; and red for dry) of globally integrated mass from the GEOS5-

AGCM REPLAY of the RPIT reanalyses (note, black and cyan curves completely overlay). This REPLAY 

did use global constraints. 

32 
with RPIT due to model updates targeted for MERRA-2, but the overall character is the same. 

The REPLAY with global constraints, as expected, shows a perfect balance between P and E. 

33 
Figure 3.13: Globally integrated evaporation and precipitation from the GMAO RPIT reanalysis and 

the two REPLAY experiments (with and without constraints). 

34 
4) Preliminary MERRA-2 Results 

The GMAO MERRA-2 was started in the spring of 2014. While not yet complete, we show here 

some preliminary results which have incorporated all updates previously discussed. The 

MERRA-2 Reanalysis is being carried out through four concurrently-running operational streams 

(initialized in January of 1979, 1991, 2000, and 2010). The following figures show results from 

each of the four streams. 

Figure 4.1 shows the time series of total surface pressure separated into its dry and wet 

components. We see that each stream is conserving its initial globally-integrated dry mass 

value of 983.24 hPa (surface pressure equivalent). The wet surface pressure component shows 

the well defined seasonal cycle associated with total water content. 

Figure 4.2 shows the time series of MERRA-2 globally integrated precipitation and evaporation 

as well as its comparison to the precipitation produced in MERRA. We see that the MERRA-2 

system has a very steady rate of precipitation compared to that from MERRA. In addition, the 

precipitation from MERRA-2 is in balance with the MERRA-2 evaporation (cf. Fig. 1.1). The 

addition of global mass and water vapor constraints within the ADAS has greatly reduced the 

sensitivity of MERRA-2 to changes in the observational database. 

Figure 4.1: Globally integrated surface pressure separated into its wet and dry components from 

the GMAO MERRA-2. 

35 
As a final preliminary examination we look at the time series of globally integrated total 

precipitable water (TPW). As indicated earlier, the globally integrated value of total water 

content is now entirely model-driven due to the constraint of vanishing globally integrated 

analysis moisture increments. In ADAS mode the magnitude of TPW is directly governed by 

model physics and only indirectly constrained by the analyzed fields (wind, temperature, 

moisture, and pressure). Therefore it is instructive to compare total water content values to an 

independent observational dataset to determine if model moisture bias too strongly influences 

total water content. For this purpose we use data from the Goddard Satellite-based Surface 

Turbulent Fluxes project (GSSTF version 2b, Shie et al. 2009) covering the period from 1987 

through 2008 measured over oceans. Figure 4.3 shows the time-series departure from GSSTF 

of total precipitable water from two streams within the MERRA-2 ADAS system covering the 

GSSTF time period and averaged over the oceanic regions consistent with GSSTF. The mean 

value of GSSTF over this period is ~2.8 g/cm2

. As a comparison we also show the total 

precipitable water departure as computed from a 30-year 1⁄2-deg AMIP simulation initialized in 

February 1979 using the MERRA-2 model as well as the results from the MERRA-1 ADAS. We 

see that while the MERRA-2 model tends to run wet in AMIP mode, the magnitude of total 

precipitable water in ADAS mode is in close agreement with the GSSTF-based data. The results 

from the MERRA-1 ADAS again show the sensitivity of the moisture variable to changes in the 

observational/satellite network. 

Figure 4.2: Globally integrated precipitation and surface evaporation from MERRA-2, and its 

comparison to MERRA precipitation (in black). 

36 
5) Summary 

In this report we have shown that recent reanalysis efforts have generated imbalances between 

globally integrated precipitation and surface evaporation. These imbalances are due to the 

observationally-based analysis adjustments, which are highly sensitive to changes in the 

observational/satellite database. In addition to this water imbalance, we have seen that there 

is a lack of correlation between trends in total surface pressure and trends in total water 

content. In fact, most of the trend in total surface pressure comes from a spurious trend in the 

dry mass. An effort has been made in this study to modify the atmospheric model, the GSI 

analysis, and the IAU assimilation procedure such that globally integrated dry mass is conserved 

and such that precipitation and surface evaporation are in model balance. Experiments have 

shown that the resulting modified algorithm conserves dry mass and substantially lessens the 

impact of changes in the observational database on precipitation. 

Figure 4.3: Globally integrated total precipitable water from MERRA-1 ADAS, 

MERRA-2 ADAS, and MERRA-2 AGCM compared with GSSTF. 

37 
Acknowledgement 

The authors thank their GMAO colleagues for discussions in group meetings throughout the 

development of this work, in particular: Stephen Bloom, Ronald Gelaro, William McCarty, and 

Andrea Molod. We also thank Atanas Trayanov for help with some implementation details. 

Experiments were carried out using the Linux cluster of the NASA Center for Climate Simulation.

6) References

Bloom, S., L. Takacs. A. DaSilva, and D. Ledvina, 1996: Data assimilation using incremental analysis 

 updates. Mon. Wea. Rev., 124, 1256-1271. 

Bosilovich, M.G., and F.R. Robertson, 2011: Global Energy and Water Budgets in MERRA. J. Climate, 

24,282-300. 

Dee, D. P., and A. M. da Silva, 2003: The choice of variable for atmospheric moisture analysis. Mon. Wea. 

Rev., 131, 155-171. 

Dee, D. P., and 35 co-authors, 2011: The ERA-Interim reanalysis: Configuration and performance of the 

data assimilation system. Quart. J. R. Meteorol. Soc., 137, 553-597. DOI: 10.1002/qj.828. 

Derber, J. C., and A. Rosati, 1989: A global oceanic data assimilation system. J. Phys. Oceanography, 19, 

1333-1347.Ebita, A., S. Kobayashi, Y. Ota, M. Moriya, R. Kumabe, K. Onogi, Y. Harada, S. Yasui, K. 

Miyaoka, K. Takahashi, H. Kamahori, C. Kobayashi, H. Endo, M. Soma, Y. Oikawa, and T. Ishimizu, 2011: 

The Japanese 55-year Reanalysis "JRA-55": an interim report, SOLA, 7, 149-152. 

El Akkraoui, A., Y. Tremolet, and R. Todling, 2013: Preconditioning of variational data assimilation and 

the use of a bi-conjugate gradient method. Quart. J. R. Meteorol. Soc., 139, 731-741. 

Hoinka, K. P., 1998: Mean global surface pressure series evaluated from ECMWF reanalysis data. Q.J.R. 

Meteorol.Soc., 124, 2291-2297. 

Holm, E. V., 2003: Revision of the ECMWF humidity analysis: Construction of a Gaussian control variable. 

Proc. Workshop on Humidity Analysis, Reading, U.K., ECMWF/GEWEX, 1-6. [Available online from 

http://old.ecmwf.int/publications/library/ecpublications/_pdf/workshop/2002/Humidity/holm.pdf] 

Kleist, D. T., D. F. Parrish and J. C. Derber and R. Treadon and W.-S. Wu and S. Lord, 2009: Introduction of 

the GSI into the NCEP’s Global Data Assimilation System. Wea. Forecasting, 24, 1691-1705. 

Lucchesi, R., 2013: File Specification for GEOS-5 FP-IT. GMAO Office Note No. 2 (Version1.2),60pp, 

available from http://gmao.gsfc.nasa.gov/pubs/office_notes. 

38 
Massart, S., B. Pajot, A. Piacentini, and O. Pannekoucke, 2010: On the merits of using 3D-FGAT 

assimilation scheme with an outer loop for atmospheric situations governed by transport. Mon. Wea. 

Rev., 138, 4509-4522. 

Reynolds, R.W., N.A. Rayner, T.M. smith, D.C. Stokes, and W. Wang, 2002: An improved in situ and 

satellite SSA analysis for climate. J. Climate, 15, 1609-1625. 

Rienecker, M.M., M.J. Suarez, R. Gelaro, R. Todling, J. Bacmeister, E. Liu, M.G. Bosilovich, S.D. Schubert, 

L. Takacs, G.-K. Kim, S. Bloom, J. Chen, D. Collins, A. Conaty, A. da Silva, et al. (2011), MERRA: NASA's 

Modern-Era Retrospective Analysis for Research and Applications. J. Climate, 24, 3624-3648, 

doi:10.1175/JCLI-D-11-00015.1. 

Saha, S., and Coauthors, 2010: The NCEP Climate Forecast System Reanalysis. Bull. Amer. Meteor. Soc., 

91, 1015–1057.doi: http://dx.doi.org/10.1175/2010BAMS3001.1

Shie, C.-L., L.S. Chiu, R. Adler, E. Nelkin, I-I. Lin, P. Xie, F.C. Wang, R. Chokngamwong, W. Olson, and D.A. 

Chu, 2009: A note on reviving the Goddard satellite-based surface turbulent fluxes (GSSTF) dataset. 

Adv. Atmos. Sci., 26(6), 1071-1080. 

Trenberth, K.E., 1991: Climate Diagnostics from Global Analyses: Conservation of Mass in ECMWF 

Analyses. J. Climate, 4, 707-722. 

Trenberth, K.E., J.R. Christy, and J.G. Olson, 1987: Global Atmospheric Mass, Surface Pressure, and 

Water Vapor Variations. J. Geophys. Res., 92, 14815-14826. 

Trenberth, K.E., and C. Guillemot, 1994: The total mass of the atmosphere. J. Geosphys. Res., 99, 23079-

23088. 

Trenberth, K.E., and L. Smith, 2005: The Mass of the Atmosphere: A Constraint on Global Analysis. J. 

Climate, 18, 864-875.

39