A new 2-D Model to analyze uncertainty sources of sparse sea surface

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1 A new -D Moel to analyze uncetainty souces of spase sea suface O patial pessue Xiang Li,3,Minhan Dai, an Guizhi Wang, oesponence to: M. Dai, mai@xmu.eu.cn.state Key Laboatoy of Maine Envionmental Science, Xiamen Univesity, Xiamen, hina,. ollege of Ocean an Eath Science, Xiamen Univesity, Xiamen, hina, 3. School of Physics, Huazhong Univesity of Science & Technology. Abstact In oe to bette compehen the global cabon cycle an peict the pognosis fo the esponse to climate change, accuate assessment of sea-ai O flux is necessay. ompaing to the elative homogeneously istibution of atmospheic O, the p O in the sea suface wate is expose to huge spatio-tempoal vaiability, which leaves a pominent uncetainty esouce. Many egional stuies typically ivie the obsevational p O ata into gi boxes so as to obtain enough ata points statistically fo thei calculatio. Howeve, using the ata insie the gi box aeas to epesent its holistic popety (such as stana eviation to epesent spatial vaiance) will mix up thee iffeent uncetainty souces. Fist, the analytical eo in the p O etemination an the associate envionmental paametes use in eiving p O. Secon, the spatial vaiance because of inhomogenous spatial patten of sea suface p O, especially the egion with a amatically ynamic cicumstance like: coastal aeas, bounaies o fonts an etc. Thi, the estimation pocess in unesampling conition, specifically, this kin of uncetainty oigins fom the pocess that using a spase ata to epesent its holistic popety of the box aea. ommon uncetainty quantification by Stana Deviation will mix up the iffeent souces of uncetainty. In this pape, it intouces an optimize poceue to etemine thee souces of uncetainty ( st analytical eo, n spatial vaiance, 3 bias fom unesampling.) using the combine emote sensing-eive an fiel-measue po ata. In oe to povie a compehensive eo assessment epot. Special Section Pacific-Asian Maginal Seas Key Points Thee souces of uncetainty of spase po ata. A new -D moel intouce in Kiging Estimation. ouple with emote sensing-eive ata to fit the -D moel. The uncetainties of spase po ata ae estimate.

2 . Intouction: Figue. Fom Global abon Buget 04 The continuing api accumulation of O in atmosphee inevitably each to an unpeceente level in human histoy, the 400 ppm mak [Global abon Buget 04]. In oe to bette compehen the global cabon cycle an peict the pognosis fo the esponse to climate change, accuate assessment of sea-ai O flux is necessay [Takahashi et al., 009, 00, 997]. Since 993, a global coopeate pogam, the DIA Ocean abon Data Management Poject, has been establishe fo collecting iscete an uneway measuement po ata. Thanks to temenous effot contibute by iniviual investigatos an goups, up till 04, moe than 9.0 million measuements of suface wate p O ata points mae ove the global oceans have been inclue in the LDEO atabase [Vesion 03]. Howeve, compaing to the elative homogeneously istibution of atmospheic O, the p O in the sea suface wate is expose to huge spatio-tempoal vaiability, which leaves a pominent uncetainty esouce. It iectly casts a shaow on the eliability of estimating the sea-ai flux. Many egional stuies typically ivie the obsevational p O ata into gi boxes so as to obtain enough ata points statistically fo thei calculation[e.g., Takahashi et al., 009; Zhai et al., 03]. Howeve, using the ata insie the gi box aeas to epesent its holistic popety (such as stana eviation to epesent spatial vaiance) will mix up thee iffeent uncetainty souces [Wang et al., 04]. Fist, the analytical eo in the p O etemination an the associate envionmental paametes use in eiving p O. Secon, the spatial vaiance because of inhomogenous spatial patten of sea suface p O, especially the egion with a amatically ynamic cicumstance like: coastal aeas, bounaies o fonts an etc [Sweeney et al., 03]. Thi, the estimation pocess in unesampling conition, specifically, this kin of uncetainty

3 oigins fom the pocess that using a spase ata to epesent its holistic popety of the box aea. Because of inhomogeneously an spasely istibution of sampling station points is common, coectly an efficiently quantifying those thee kins of uncetainties is stongly equie. This stuy bases on the spatial analysis an Kiging estimation, using a new two-imensional spatial coelation moel to quantify the contibution of the spatial vaiance an unesampling uncetainty in sea suface po ata an futhe to optimize the assessment of sea-ai O flux.. Analytical Eo (Em). Spatial Vaiance 3. The isk fom estimation in unesampling conition. Given that its necessity when evaluating iffeent stuies, ou appoach shall have wiely applications as an uncetainty analysis tools.. p O atabase an methoology to quantify the uncetainty of spase ata. Monitoing the vaiability of sea suface p O, people use ships an othe platfoms geneates lage amounts of ata fom heteogenous souces. Since 993, a global coopeate pogam via a numbe of U.S. an intenational ocean-obseving pogams, the DIA Ocean abon Data Management Poject has been establishe fo collection of iscete an uneway measuements po ata. The atabase use in this stuy ae fom an uneway measuements in ES August, 009 an Global Ocean Suface Wate Patial Pessue of O Database (Vesion 03). Figue. Flowchat of the quantifying uncetainty fo po ata. RS- po epesents emote sensing-eive po ata, an uneway epesents fiel-measue po ata. Fo a given sea suface po fiel ata set, the analytical eo, Em, is nomally calculate by aggegating all the eos intouce in the measuement an ata euction of po in many stuies[]. In this stuy, it will focus on a stana poceue fo estimating the est two souces of uncetainty, the spatial vaiance an unesamping. An this methoology will be moe efficient with ai of a concuent high-spatial-esolution satellite-eive po ata set fom [Bai et al., 05] in the same stuying egion. But, it is not equie so. The potocol of this metho will be summaize in a simplifie way in Figue.. Fist, using spatial ata (Uneway o Remote-Sensing measuements) to o the spatial analysis fo estimating a new two-imensional spatial coelation

4 moel. Secon, combining the obtaine statistical moel an uneway measuements to quantify the coelation of each ata pai. Then, basing on Oinay Kiging (OK) Estimation to povie a full coveage ata set of the stuy egion. Finally, estimate the spatial vaiance via calculating the stana eviation of the full coveage ata set an the unesampling uncetainty fom the estimation vaiance of OK. The etails will be as the following steps.. Spatial analysis an cicula segment moel. Fo a given spatial ata set, such as ES August 009, nomally they have an inheent popety, the spatial epenence in attibute values, which means the values fo the same attibute measue at locations that ae nea to one anothe ten to be simila, an ten to be moe simila than values sepaate by lage istances. In this stuy, in oe to povie an efficient an simple algoithm, we assume the epenency stuctue is also isotopic, the same on both axes, which is an ieal conition. Although this assumption oes not sufficiently consie the ocean system, which is pesistently ynamic in physical, chemical an biological pocess, it eflects some basic infomation o popety fom ata itself. The paticula steps to quantifying the stuctue of spatial epenency in a ata set will be fully iscusse in [Robet Haining, 993]. ˆ( h) [(/ N( h)) ( z( i) z( j))] z( i) z( j) i j [( i, j) ( i, j) h)] Whee, (h) is the estimate of spatial covaiance at istance h. Figue. is a conceptual example of a typical semi-vaiogam fo the case whee spatial epenence in attibute values. This spatial autocoelation analysis is the basis fo the next estimation. () Figue.3 (onceptual moel fo an autocovaiance function (h)). i Fom the above spatial coelation analysis, the next step is fitting statistical moel. onsieing ealing with a -imensional poblem, we popose a new -D statistical moel as shown in Figue.3. This moel epesents the coelation of two ata points by calculating the segment aea of them, also in the case of an isotopic weakly stationay pocesses. This means that any pemissable covaiance function also can be use as a moel fo the semi-vaiogam of a weakly stationay pocess, the etaile explanation ae given by [essie, 99, p87].

5 Figue.4 The -imensional covaiance statistical moel. Whee the ovelap of two ciculas at a istance will be calculate by the flowing fomula: ) ( acsin 4 A ovelaping () The covaiance function of this D moel is: icula Moel: a h a h h h 0 0 ] acsin 4 [ 0 ) ( 0 (3) The sill is the plateau a function eaches an which coespons to (0). The ange is the istance at which the plateau is eache. Detail see in [Robet Haining, 993, p95].. Deiving fom Kiging Estimation. Afte finishing the spatial analysis an moel fitting, the thee constants of statistical moel, icula Moel, will be known. An using this obtaine moel, we coul statistically quantifying each ata pai s coelation on aveage. Then, eiving the equations fom oinay kiging with two conitions in tems of D covaiance function. This peiction is the homogeneously linea combination of the ata set that minizizes the mean squae peiction eo an will be fully iscusse in [essie, 99].

6 n n nn 0 n 0 0n (4) The above equation set is to calculate a single estimation point, eive by us following Kiging oinay metho. An the esults will be the following: Estimation of a single point Z0. Uncetainty of OK estimation that bings in. Z n * 0 i i i Z( x ) (5) n ( 0) i(x 0, x i ) OK (6) i Using this metho, povie a full coveage estimation of spase ata set. Then, using N s ( Z i Z ), (7) N i to estimate the spatial vaiance via calculating the stana eviation of the full coveage ata set an the unesampling uncetainty fom the estimation vaiance of OK. n ( 0) i(x 0, x i ) u (8) i Finally, integating the thee uncetainty, the analytical eo, the spatial vaiance an the unesampling vaiance, an in this case, thee of them all ae totally inepenent. The total uncetainty () is: T Em s u (9) 3. Applications of the Uncetainty Quantification to iffeent cases. Fom above, we intouce a moifie Kiging estimation metho. This metho of quantifying uncetainty was applie to iffeent case stuies, such as, the uneway suface po ata fom the East hina Sea collecte in August 009, shown in Figue. The stana poceue will be like, spatial analysis (Figue.4), estimation of -D statistical moel an estimation of full coveage ata (Fig.5.a) an its uncetainty (Fig.5.b).

Figue.4 (a) Uneway suface po ata fom the East hina Sea collecte in August 009 fom [Wang et al.

5 (a) A full coveage estimation of ES August, 009 ata set.

While calculating this ata set o othe case stuies of a fiel tip.

7 Figue.4 (a) Uneway suface po ata fom the East hina Sea collecte in August 009 fom [Wang et al., 04 ], (b)spatial covaiance function to this ata set. 3. Results Valiation (ES August, 009) Figue.5 (a) A full coveage estimation of ES August, 009 ata set. (b) The unesampling uncetainty istibution of each estimation points. While calculating this ata set o othe case stuies of a fiel tip. Thee ae always some locations of ata stations having the same cooinates, which means moe than one measuement at a single location. An it will cast a citical poblem fo this OK metho, because the equation set () is singula, not full ank, so that they can not be solve. In this stuy, we have use a tick to eal with this poblem, anomly isplacing each ata station a vey little bit in space, that the isplacement is tenth lowe the minimum esolution of uneway measuement. Howeve, it is still nee moe stuies to evaluate how much uncetainty will be taken in ue to this ough tactic.

Meanwhile, we have a concuent high-spatial-esolution satellite-eive po ata set fom [Bai et al., 05] in the same stuying egion shown in Figue.6.

An most impotantly, the istibution of unesampling uncetainty was stongly popotional to the ensity of uneway ata, which is quite ecent an logically.

An fo each estimation points, the uncetainty will ecease when having some ajacent ata points.

Fom the cuent huge amount of po atabase, Aiming to avoi the lagely tempoal vaiability, we only picke up some fiel tips within a shot time peio, nomally

(b) The autocovaiance function of this ata set. Fom above, it showe an inconfomity between the auto-coelation function an ieal moel.

8 Meanwhile, we have a concuent high-spatial-esolution satellite-eive po ata set fom [Bai et al., 05] in the same stuying egion shown in Figue.6. We can see a pominent coheence between estimations esults (Figue.5.a) an Remote-Sensing ata (Figue.6). An most impotantly, the istibution of unesampling uncetainty was stongly popotional to the ensity of uneway ata, which is quite ecent an logically. Hee, in this metho, it uses the sum of the squae eviation, N times vaiance, as the maxim of the estimation uncetainty. An fo each estimation points, the uncetainty will ecease when having some ajacent ata points. Basing on coelate ata an infomation will euce the isks of estimation, o specifically, uncetainty. Figue Othe ase Stuies- If Heteogeneity? Fom the cuent huge amount of po atabase, Aiming to avoi the lagely tempoal vaiability, we only picke up some fiel tips within a shot time peio, nomally less than half a month. Two typical cases will be iscusse in the following. Figue.7 (a).uneway measuement of uise: Bal Jan 995, ata fom DIA. (b) The autocovaiance function of this ata set. Fom above, it showe an inconfomity between the auto-coelation function an ieal moel. So as the esult, it showe a poo quality of not only the full coveage of estimation but also the uncetainty of estimation points. So, to pomote the quality of esults, we ecommen to geogaphically efine subsets of ata set if heteogeneity is suspecte as the uneway measuement showe, the obsevation coveage of this fiel tip can not fully etect its spatial stuctue.

3.3. If Ovelapping? Figue.8 (a) Uneway measuement of uise: GG08 May 008, ata fom DIA. (b) The full coveage of estimation. (c) The uncetainty of estimation. As shown in Figue.

This will esult in a phenomenon calle nugget effect an also weaken the efficiency of this metho. 4.

It successfully povies a full coveage of estimation an its uncetainty fo calculating the spatial vaiance an unesampling uncetainty of stuying egion.

9 3.3. If Ovelapping? Figue.8 (a) Uneway measuement of uise: GG08 May 008, ata fom DIA. (b) The full coveage of estimation. (c) The uncetainty of estimation. As shown in Figue.8, hee is an anothe typical status with confomity between the auto-coelation function an ieal moel. But, some ata stations have ovelappe. This will esult in a phenomenon calle nugget effect an also weaken the efficiency of this metho. 4. Summay Fom all above, we intouce a geneal metho to calculate the thee uncetainties of spatial spase ata, such as ou focus, the sea suface po ata. It successfully povies a full coveage of estimation an its uncetainty fo calculating the spatial vaiance an unesampling uncetainty of stuying egion. An if with ai of a concuent high-spatial-esolution satellite-eive po ata set, this metho will povie a moe obust esult. Howeve, the application of this metho to some fiel tip measuement fom DIA atabase still eveal some citical weakness in seveal typical conitions, fo instance, heteogeneity an ovelapping. onsieing this stuy will have a wiespea application in the futue, thee ae still some optimizing wok nee to be one.

10 Refeences: Bai, Y., W.-J. ai, X. He, W. Zhai, D. Pan, M. Dai, an P. Yu (05), A mechanistic semi-analytical metho fo emotely sensing sea suface po in ive-ominate coastal oceans: A case stuy fom the East hina Sea, Jounal of Geophysical Reseach: Oceans, 0(3), essie, N. (99), STATISTIS FOR SPATIAL DATA, Tea Nova, 4(5), 63&nash;67. essie, N., an N. H. han (989), Spatial Moeling of Regional Vaiables, Jounal of the Ameican Statistical Association, 84(406), Haining, R. (003), Spatial Data Analysis, Spatial Data Analysis, by Robet Haining, pp. 45. ISBN ambige, UK: ambige Univesity Pess, June 003., (), 45. Le Quéé,., et al. (05), Global cabon buget 04, Eath System Science Data, 7(), Legene, P. (993), Spatial autocoelation: touble o a new paaigm, Ecology, 74(6), Legene, P., N. L. Oen, R. R. Sokal, A. Vauo, an J. Kim (990), Appoximate analysis of vaiance of spatially autocoelate egional ata, Jounal of lassification, 7(), Shen, S. S. P., H. Yin, an T. M. Smith (007), An Estimate of the Sampling Eo Vaiance of the Gie GHN Monthly Suface Ai Tempeatue Data, Jounal of limate, 0(0), Shen, S. S. P.,. K. Lee, an J. Lawimoe (0), Uncetainties, Tens, an Hottest an olest Yeas of U.S. Suface Ai Tempeatue since 895: An Upate Base on the USHN V TOB Data, Jounal of limate, 5(), T, T., F. RA, W. RF, W. RH,. DW, S. S, an T. TT (997), Global ai-sea flux of O : An estimate base on measuements of sea ai po iffeence, Poceeings of the National Acaemy of Sciences, 94(6), Takahashi, T., S.. Suthelan, R. Wanninkhof,. Sweeney, R. A. Feely, D. W. hipman, B. Hales, G. Fieeich, F. havez, an. Sabine (009), limatological mean an ecaal change in suface ocean po, an net sea ai O flux ove the global oceans, Deep Sea Reseach Pat II: Topical Stuies in Oceanogaphy, 56(), Takahashi, T., et al. (00), Global sea ai O flux base on climatological suface ocean p O, an seasonal biological an tempeatue effects, Deep Sea Reseach Pat II: Topical Stuies in Oceanogaphy, 49(9 0), Tans, P. P., I. Y. Fung, an T. Takahashi (990), Obsevational contains on the global atmospheic O buget, Science, 47(4949), Taylo, J. R. (997), An intouction to Eo Analysis, Univesity Science Books, 56(4), Wang, G., M. Dai, S. S. P. Shen, Y. Bai, an Y. Xu (04), Quantifying uncetainty souces in the gie ata of sea sufaceo patial pessue, Jounal of Geophysical Reseach: Oceans, 9(8), Wanninkhof, R., G. Pak, an T. Takahashi (0), Global ocean cabon uptake: magnitue, vaiability an tens, Biogeosciences Discussions, 0(8),

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