Efficacy of Tendency and Linear Inverse Models to Predict

INTERNATIONAL JOURNAL OF CLIMATOLOGYInt. J. Climatol. (2018)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/joc.5442

Efficacy of tendency and linear inverse models to predictsouthern Peru’s rainy season precipitation

Shu Wu,a* Michael Notaro,a Stephen Vavrus,a Eric Mortensen,b Rob Montgomery,c

José de Piérolad and Paul Blockb

a Nelson Institute Center for Climatic Research, University of Wisconsin-Madison, WI, USAb Department of Civil and Environmental Engineering, University of Wisconsin-Madison, WI, USA

c Montgomery Associates Resource Solutions LLC, Cottage Grove, WI, USAd Southern Peru Copper Corporation, Santiago de Surco, Lima, Peru

ABSTRACT: Southern Peru receives over 60% of its annual climatological precipitation during the short period ofJanuary–March. This rainy season precipitation exhibits strong inter-annual and decadal variability, including severe droughtevents that incur devastating societal impacts and cause agricultural communities and mining facilities to compete for limitedwater resources. Improving existing seasonal prediction models of summertime precipitation could aid in water resourceplanning and allocation across this water-limited region. While various underlying mechanisms modulating inter-annualvariability have been proposed by past studies, operational forecasts continue to be largely based on rudimentary ElNiño-Southern Oscillation (ENSO)-based indices, such as Niño3.4, justifying further exploration of predictive skill. To bridgethe gap between understanding precipitation mechanisms and operational forecasts, we perform systematic studies on thepredictability and prediction skill of southern Peru’s rainy season precipitation by constructing statistical forecast modelsusing best available weather station and reanalysis data sets. We construct a simple regression model, based on the principalcomponent (PC) tendency of tropical Pacific sea surface temperatures (SST), and a more advanced linear inverse model (LIM),based on the empirical orthogonal functions of tropical Pacific SST and large-scale atmospheric variables from reanalysis. Ourresults indicate that both the PC tendency and LIM models consistently outperform the ENSO-only based regression models inpredicting precipitation at both the regional scale and for individual station, with improvements for individual stations rangingfrom 10 to over 200%. These encouraging results are likely to foster further development of operational precipitation forecastsfor southern Peru.

KEY WORDS southern Peru precipitation; operational forecast; tendency model; linear inverse model; statistical forecast;drought

Received 18 October 2017; Revised 2 January 2018; Accepted 4 January 2018

1. Introduction

Southern Peru’s complex topography, with elevationsvarying from hundreds to thousands of meters from thewestern coast to the eastern plateau, leads to uniquefeatures of local weather and climate. Even though it isadjacent to the eastern Pacific Ocean, little water vapour(<8%) (Perry et al., 2014) is routinely transported fromthe west coast to this semi-arid area due to a stable atmo-spheric boundary layer over the cold ocean (Garreaudet al., 2003; Mechoso et al., 2014) although at times,in particular in winter, the northwards displacements orcut-offs of cold air-masses from the Pacific may resultin some snowfall (Vuille and Ammann, 1997). In fact,the limited moisture reaching southern Peru is mainlytransported from the lowlands on the east side of theAndes mountains through upslope winds, which are pre-dominantly affected by the upper-level local zonal wind

* Correspondence to: S. Wu, Center for Climatic Research, 1225 W.Dayton St., Madison, WI 53706, USA. E-mail: [email protected]

anomalies, with easterly winds favouring wet conditionsand westerly winds favouring dry conditions in southernPeru (Garreaud, 1999; Garreaud et al., 2003; Perry et al.,2014). The northerly wind may also contribute someportion of the region’s moisture transport according totrajectory analyses (Perry et al., 2014, 2017), but nosignificant correlation has been identified between themeridional wind anomalies and local precipitation in ourstudy.

The main rainy season covers January–March (JFM),which accounts for approximately 60% of the total annualprecipitation; however, strong inter-annual variability inprecipitation also exists. In 2016, the most recent droughtyear, the total precipitation received in southern Peru (seeSection 2 for the station information) during JFM wasapproximately 200 mm, or about 30% below the long-termseasonal mean, resulting in agricultural losses up to 75%and severe local economic decline (Autoridad Nacionalde Agua (ANA), 2016). Various mechanisms are believedto control JFM precipitation variability (Garreaud et al.,2003). At the intra-seasonal timescale, variability is mainly

© 2018 Royal Meteorological Society

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Table 1. Station information across southern Peru, together with mean annual precipitation and the percentage of annual precipitationfalling during JFM.

Station number Station name Latitude Longitude Elevation(m)

Annualprecipitation (mm)

JFM percentageof annual (%)

1 Toquepala mina 17.26∘S 70.59∘W 3400 81 832 Quebrada honda 17.06∘S 70.55∘W 3995 257 843 Tacalaya 17.06∘S 70.41∘W 4415 438 744 Sucches lt 16.94∘S 70.39∘W 4468 390 725 Pasto grande 16.72∘S 70.23∘W 4546 487 676 Cuajone mina 17.04∘S 70.72∘W 3576 137 837 Umalso 16.87∘S 70.42∘W 4609 415 728 Calacoa 16.73∘S 70.68∘W 3478 413 869 Susapaya 17.35∘S 70.03∘W 3309 214 8410 Sttajara 17.35∘S 70.13∘W 3166 127 8511 Talabaya 17.55∘S 69.99∘W 3409 194 8612 Toquela 17.64∘S 69.94∘W 3445 169 8613 Challapalca 17.23∘S 69.78∘W 4190 410 6914 Chivay 15.64∘S 71.60∘W 3633 424 6915 Imata 15.84∘S 71.09∘W 4519 536 6816 Cabanaconde 15.62∘S 71.97∘W 3379 405 7817 Salamanca 15.50∘S 72.83∘W 3203 342 7818 Ubinas 16.38∘S 70.86∘W 3370 310 7519 Madrigal 15.61∘S 71.81∘W 3262 427 7420 Crucero alto 15.80∘S 70.91∘W 4470 589 6521 Pillones 15.98∘S 71.21∘W 4310 414 6922 Angostura 15.17∘S 71.63∘W 4150 800 6423 Puno 15.83∘S 70.01∘W 3820 722 6024 Cabanillas 15.64∘S 70.35∘W 3919 664 5825 Capazo 17.19∘S 69.74∘W 4530 534 7126 Desaguadero 16.57∘S 69.04∘W 3860 717 6327 Isla Taquile 15.77∘S 69.69∘W 3850 1213 5828 Laraqueri 16.15∘S 70.07∘W 3900 754 6129 Mazo cruz 16.74∘S 69.71∘W 4100 544 66

affected by the position and intensity of the Bolivian High,which is an upper-level (over 300 hPa) anticyclone thatdevelops during the boreal summer over the Bolivian Alti-plano. Its position is mainly determined by the convec-tive heating due to the Amazonian precipitation (Lentersand Cook, 1997). A weakened and northwards-displacedBolivian High is often associated with persistent dryness,and vice versa. Additionally, as with many other phenom-ena in the climate system, southern Peru precipitation ischiefly affected by tropical Pacific sea surface tempera-ture (SST) anomalies, namely the El Niño-Southern Oscil-lation (ENSO) (Ropelewski and Halpert, 1987; Masonand Goddard, 2001). At the seasonal timescale, precip-itation variability is mainly controlled by the meridion-ally directed seasonal migration of upper tropospherewinds. The expansion of the equatorial easterly windsover southern Peru leads to the onset of the rainy season.Thus, the primary driver of precipitation variability acrosstimescales is through the modulation of local zonal windanomalies and associated moisture transport (Vuille, 1999;Garreaud and Aceituno, 2001).

Current forecasting skill of precipitation anomalies inthis topographically complex region is limited. Availablereal-time forecasts from dynamical models (see Section 3for details) tend to predict dry conditions for most years.

Statistical prediction models based on simple Niño1+2and Niño3.4 indices (Trenberth, 1997) have been proposed(e.g. Lagos et al., 2008; Cid-Serrano et al., 2015) to pre-dict Peru precipitation; however, there are years when thetwo indices have different signs, leading to a selectiondilemma, or instances of when the indices agree in signbut predict precipitation anomalies in the wrong direc-tion. Given the limitations of current seasonal forecast sys-tems for this region, we evaluate in situ observations andatmospheric and oceanic reanalysis data to gauge potentialimprovements in season-ahead prediction skill. We exam-ine forecast skill at both the regional scale and for individ-ual stations.

The manuscript is organized as follows. In Section2, we highlight characteristics of southern Peru rainfalland its relationship with ENSO based on previouslyunanalysed rain gauge data. In Section 3, we review thestatus of available operational and statistical precipitationforecasts for the region. We then introduce a simple prin-cipal component (PC) tendency model in Section 4 thataccounts for the temporal evolution of tropical Pacific SSTanomalies, which has the potential to clarify the ENSOprecipitation relationship. In Section 5, we introduce amore complex statistical model, the linear inverse model(LIM), particularly to address predictions at the level of

© 2018 Royal Meteorological Society Int. J. Climatol. (2018)

TENDENCY AND LIM MODEL FOR PERU PRECIPITATION FORECAST

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ocean and lakes. [Colour figure can be viewed at wileyonlinelibrary.com].

individual stations. Section 6 contains the summary anddiscussion.

2. Southern Peru precipitation characteristicsand their relationship with ENSO

Daily precipitation gauge observations from 29 stations(Table 1) are provided by Peru’s national meteorolog-ical and hydrological agency, the Servicio Nacionalde Meteorología e Hidrología del Peru (SENAMHI),and Southern Peru Copper Corporation (SPCC). Thedata set covers the period of 1966–2017 and containsless than 10% missing values, which were filled usingmultiple linear regression models based on monthlystatistics (Mortensen et al., 2018). National Centers forEnvironmental Prediction (NCEP)–National Center forAtmospheric Research (NCAR) Reanalysis atmosphere

data (Kalnay et al., 1996) and Hadley Centre Sea Iceand Sea Surface Temperature (HADISST) (Rayner et al.,2003) are also explored in the article.

Southern Peru (15∘–18∘S, 73∘–68∘W) is located at thecentre of the Altiplano, which is surrounded by com-plex cordilleras. In general, annual mean precipitationdecreases from the northeast to the southwest across thestudy region (Figure 1(a) and Table 1), as moisture istransported across the eastern slope via upslope winds(Garreaud et al., 2003). JFM precipitation accounts for amuch larger percentage of annual precipitation in the west-ern study region than in the east (Figure 1(b) and Table 1),indicating that the rainy season in the west is much moreconcentrated during JFM.

In order to understand regional variability and poten-tial predictability, we define a domain-averaged seasonal(JFM) southern Peru precipitation index (SPPidx) based


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line indicates the mean value. [Colour figure can be viewed at wileyonlinelibrary.com].

on the average precipitation across the 29 stations, produc-ing a climatological mean of approximately 300 mm. TheSPPidx well represents the precipitation anomalies of moststations (Figure 2) in which most stations show positive(negative) precipitation anomalies when SPPidx is above(below) normal, with an average percentage of sign con-sistence around 80% (Figure 2(c)). There are years, how-ever, when several stations exhibit substantial precipitationanomalies of the opposite sign; this inter-station differenceis likely due to atmospheric interactions with the com-plex topography, as further discussed in Section 5. Markedinter-annual variability is evident (Figures 2(a) and (b)),with severe drought years (1983 and 1992) and wet years(1973, 1984, and 2001). Decadal variability is also evident,with notable wet and dry periods during 1967–1973 and1987–1998, respectively.

To confirm that the inter-annual variability of Alti-plano precipitation is predominantly associated with localzonal wind anomalies, we define a zonal wind index(UWNDidx) based on area-average 200-hPa zonal windanomalies over southern Peru (75∘–65∘W, 15∘–25∘S).UWNDidx exhibits a strong negative temporal correlationof −0.77 with SPPidx (Figure 3(a)), which is significant at95% confidence level. Indeed, all severe drought and wetevents are related to westerly or easterly wind anomalies,respectively, which are consistent with previous findings(Garreaud et al., 2003). Furthermore, we find that thezonal wind anomalies are associated with Pacific SSTanomalies (correlation R= 0.71; Figure 3(b)), which notsurprisingly results in a significant correlation betweenthe Niño3.4 index and SPPidx (R=−0.56; Figure 3(c)).ENSO’s impact on the overlying atmosphere over southern

Peru is likely triggered by anomalous condensational heat-ing of the troposphere over the tropical Pacific Ocean,which consequently excites the eastwards propagationof atmospheric Kelvin waves (Matsuno, 1966; Gill,1980) and causes zonal wind anomalies to establish oversouthern Peru.

3. Current skill of operational precipitation forecasts

Several sources of real-time seasonal forecasts of globalprecipitation based on multi-coupled dynamical climatemodels are available, including the National Multi-ModelEnsemble (NMME) (Kirtman et al., 2014), InternationalResearch Institute for Climate and Society (IRI) (Barn-ston et al., 2010), and Asia-Pacific Economic Coopera-tion Climate Center (APCC). The deterministic, real-timeforecasts from these three sources for the past 10 years(Table 2) tend to be dry biased, inaccurately predicting dryconditions for most years. Although rarely predicting wetconditions, the models usually produce accurate forecastsin actual wet years, indicating that the models are ratherconservative in forecasting wet events.

Overall, dynamic model forecasts tend to have dramati-cally dry biases.

In contrast to dynamical model results, simpleENSO-based statistical regression models, as currentlyused operationally by SPCC and other organizations inthe region, produce superior results, with the Niño3.4index outperforming Niño1+2 (Figures 3(c), 4(a), and(b)). Although the forecast errors from each regressionmodel are highly correlated with each other (R= 0.92),




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Figure 3. Scatter plots comparing normalized variables, including (a) SPPidx in JFM versus UWNDidx in JFM, (b) UWNDidx in JFM versus Niño3.4index in the previous December, and (c) SPPidx in JFM versus Niño3.4 index in the previous December during 1966–2016. [Colour figure can be

viewed at wileyonlinelibrary.com].

the Niño3.4 model has a smaller overall forecast errorand more often correctly predicts the sign of the anomaly(30 years with Niño3.4 vs 22 years with Niño1+2 across52 analysis years; Figure 4(b)). This was the case forthe JFM 2017 rainy season, in which the Niño1+2 indexindicated anomalously warm waters in the eastern tropicalPacific and Niño3.4 index indicated anomalously coldwaters in the central tropical Pacific during the preced-ing December 2016. Even in years when Niño1+2 andNiño3.4 anomalies have the same sign, their use may stilllead to inaccurate forecasts. For example, the JFM 1973rainy season, which followed a severe El Niño event in1972, was expected to be dry based on the aforementionedNiño indices, yet it turned out to be extremely wet insouthern Peru.

Previous studies (e.g. Vuille et al., 2000a, 2000b; Ron-chail et al., 2002, 2005) suggest that SST anomalies acrossocean basins other than the tropical Pacific may also affectsouthern Peru precipitation; thus, we examine the forecastskill of regression models based on adding various oceanicindices as additional predictors beyond the Niño3.4 index(Figure 5). At first, the temporal correlation forecast skillbased on individual indices, such as Niño1+2, Niño3.4,Niño4 (Trenberth, 1997), North Tropical Pacific (NTP)index (see the figure caption for domain area), SouthTropical Pacific (STP) index, North Tropical Atlantic(NTA) index, South Tropical Atlantic (STA) index, Trop-ical Pacific Meridional Gradient index (NTP-STP), Trop-ical Atlantic Meridional Gradient index (NTA-STA), andNorth Tropical index (NTP+NTA), is examined. Of this



S. WU et al.

Table 2. SPPidx versus real-time forecast results from IRI,APCC, and NMME.

SPPidx IRI APCC NMME

2016 − − − − − − −2015 + − − −2014 − − − − N/A2013 + − − −2012 ++ − + N/A2011 + + ++ N/A2010 − − − − N/A2009 − − − N/A2008 − − − N/A2007 + − N/A N/A

‘+’ and ‘−’ represent anomalous wet and dry conditions, and ‘++’and ‘− −’ represent extremely (±1 standard deviation) wet and dryconditions, respectively. ‘N/A’ means no available information or nosignificant value. It should be noted that IRI’s forecast tends to mask outthe southern Peru region, so the forecast result from the closest availablepoint is chosen as the forecast result for southern Peru. The deterministicmulti-model mean forecasts from APCC and NMME are selected asreal-time forecast results from these centres.

set, the most skilful predictor is the Niño3.4 index. Othertropical Pacific indices also exhibit some skill. In contrast,little correlation is identified for tropical Atlantic indices.Multiple regression forecasts, in which Niño3.4 index iscombined with another index as predictors, also show nosignificant improvement over the simple Niño3.4-basedunivariate regression model.

Overall, the current skill of operational precipitationforecasts in southern Peru is still limited. The best modelis Niño3.4 index based regression model. In the follow-ing section, we will use the Niño3.4 index model as thebenchmark for developing and evaluating new forecastmodels.

4. PC tendency model

Recent studies have revealed that ENSO may actuallybe expressed through different ‘flavours’ (Johnson, 2013;Capotondi et al., 2015), such as the eastern El Niño andcentral El Niño (‘Modoki’) (Ashok et al., 2007; Kao andYu, 2009). Depending on the season and location ofmaximum SST anomalies in the tropical Pacific Ocean,the impacts of these two different flavours of ENSO onregions such as Japan, New Zealand, and west coast ofthe United States can be opposite. Perhaps not surprisinglyas well, each type also has varying effects on Peruvianprecipitation (Sulca et al., 2017), likely due to differ-ing responses of the overlying atmospheric circulation.A single Niño index, as applied in existing operationalmodels, is therefore unlikely to sufficiently represent suchspatial variations in tropical SST anomalies.

To account for this diversity in SST anomalies, we evalu-ate the spatio-temporal evolution of the first PC (PC1 here-after) of monthly SST anomalies across the whole tropicalPacific Ocean. The differences between each monthly pairof PC1s (e.g. December–November) indicate the direc-tion or tendency of SST anomalies over time, namely thestrengthening or weakening of ENSO events. The advan-tage of adding tendency terms as predictors imbedded inthe physical–empirical model has been showed to be avery promising method to improve forecasts of Indian andEast Asian monsoon rainfall (Wang et al., 2015; Li andWang, 2016; Li and Wang, 2017; Li et al., 2017; Zhu andLi, 2017). Because ENSO forecast skill drops significantlyprior to May due to the spring forecast barrier (e.g. Bal-maseda et al., 1995; Duan and Wei, 2013), we explorePC tendencies from June to December to identify poten-tially skilful predictors for subsequent JFM precipitationin southern Peru.

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Figure 4. Scatter plots of (a) normalized SPPidx in JFM versus Niño1+2 index in the previous December, (b) forecast error (predicted minus observednormalized SPPdix) of Niño1+2 versus Niño3.4. Niño3.4 provides better forecast skill in 30 years (green/light dots) and worse skill in 21 years

(blue/dark dots) than Niño1+2. [Colour figure can be viewed at wileyonlinelibrary.com].




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Figure 5. Temporal correlation forecast skill for SPPidx based on single index models (blue/dark bars) and multiple regression models (yellow/lightbars), based on Niño3.4 plus an additional predictor. Indices are defined as domain-averaged SST anomalies across the following regions: Niño1+2(90∘–80∘W, 0–10∘S), Niño3.4 (150∘–120∘W, 5∘S–5∘N), Niño4 (160∘E–150∘W, 5∘S–5∘N), North Tropical Pacific (NTP) index (120∘–290∘E,0–20∘N); South Tropical Pacific (STP) index (120∘–290∘E, 20∘S–0); North Tropical Atlantic (NTA) index (300∘–360∘E, 0–20∘N); South TropicalAtlantic (STA) index (300∘–360∘E, 20∘S–0); NTP-STP (Tropical Pacific Meridional Gradient); NTA-STA (Tropical Atlantic Meridional Gradient);

and NTA+NTP (North Tropical index). [Colour figure can be viewed at wileyonlinelibrary.com].

The spatial patterns related to the PC1s, namelythe empirical orthogonal functions (EOFs) for June–December, clearly reflect the evolution of a typical ENSOevent, with SST anomalies first evident along the west-ern coast of Peru in June and gradually expanding tothe central Pacific Ocean (Figure 6) (Rasmussen andCarpenter, 1982). The amplitude and explained vari-ance also increase with time, reaching a maximum inDecember.

The PC tendency index was subsequently correlatedwith SPPidx (Table 3) to understand how the relationshipvaries with different time lags. The November–AugustPC1 tendency correlates most highly with SPPidx. Thelinear regression framework based on this new predictorcan be written as follows:

SPPidxpred = 𝛼∗ (PC1NOV − PC1AUG

)+ 𝛽

where 𝛼 and 𝛽 are coefficients for linear regressionmodel. For a centralized time series, 𝛽 = 0, and 𝛼 canbe estimated based on the least squares method. Usingthis tendency approach, drop-one cross-validated pre-dictions of SPPidx, in which data from a given year aredropped and re-predicted using coefficients estimatedbased on the remaining data, improve by nearly 20%(R= 0.65), as compared with the simple Niño3.4-basedmodel (Figure 7).

The reason why the PC tendency model outperformsthe simple Niño3.4 model is subtle yet important. Whenexploring the related circulation and moisture fields forthe two models, based on correlating the Niño3.4 index,PC tendency index, or SPPidx with global environmen-tal variables, such as SST or wind fields, the large-scale

correlation patterns are generally quite similar (figures notshown), due to the high correlation (∼0.7) between the PCtendency index and Niño3.4 index. However, one featurethat stands out in a correlation map between 500-hPa rela-tive humidity and the PC tendency index (Figure 8(b)) is aclear seesaw dipole structure between the Bolivian Highand Brazil Low (Lenters and Cook, 1997). This is alsoevident in a correlation map with SPPidx (Figure 8(c)),but is much less distinct in a correlation map with theNiño3.4 index (Figure 8(a)). This dipole structure has beenshown to be the major teleconnection pattern in SouthAmerica during dry events in the central Altiplano (Sulcaet al., 2016). Its formation is ultimately due to the con-vective heating related to inland Amazonian precipita-tion (Lenders and Cook, 1997). The improvement of thePC tendency model is likely due to better representa-tion of this regional teleconnection pattern. Additionally,the moisture anomalies over the Indian Ocean and theArabian Sea are also well captured by the PC tendencymodel, which may further contribute to better forecastskill through their impacts on Madden–Julian Oscillation(MJO) (Zhao et al., 2013), and consequently affect SouthAmerican (Alvarez et al., 2016; Shimizu and Ambrizzi,2016). The PC tendency model forecast can be issued atthe end of November to provide 1 month longer lead timethan the December-based Niño3.4 index model, which isan additional advantage.

5. LIM model

Although the simple index models show moderate skill inforecasting the domain-averaged precipitation index, they



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Figure 6. Tropical Pacific SST EOF1 for each calendar month during June–December. Thick solid lines represent “0”. Thin solid lines representpositive values. Thin dashed lines represent negative values. [Colour figure can be viewed at wileyonlinelibrary.com].




Table 3. Temporal correlation coefficients between PC tenden-cies and SPPidx.

Beginningmonths

Ending months

Jul Aug Sep Oct Nov Dec

Jun 0.200 0.271 0.468 0.561 0.592 0.603Jul – 0.248 0.494 0.594 0.624 0.625Aug – – 0.566 0.653 0.665 0.656Sep – – – 0.579 0.588 0.573Oct – – – – 0.399 0.396Nov – – – – – 0.245

PC tendencies is defined as PCs of ending months minus PCs of begin-ning months. Significant values at 95% confidence level are bolded. Val-ues bigger than 0.6 are in italics.

are not particularly well-suited for predicting precipitationat individual stations (Figure 9). Forecasting individualstations’ precipitation is typically more challengingthan forecasting regional averages, due to small-scaleprecipitation heterogeneity and greater stochasticity. Inorder to explore the predictability of precipitation atindividual southern Peruvian stations, we adopted a LIMmodelling approach, as widely used by the NationalOceanic and Atmospheric Administration for climateforecast (NOAA) (Newman et al., 2009, 2013). Only abrief introduction is provided in this article. Details canbe found in Penland (1989) and Penland and Magorian(1993).

LIM assumes that a system can be represented by twomajor components, namely a linear part and a randompart. Any nonlinear contribution is represented by therandom component. LIM can be summarized as a linearstochastic differential equation (Penland, 1989; Penland

and Magorian, 1993; Alexander et al., 2008), such that:

dxdt

= Bx + 𝝃

where x is the state variable, 𝝃 is the random noise,and coefficients matrix B is the evolution operator.The name ‘inverse’ comes from the fact that matrixB and random noise statistics, such as the amplitudeand spatial pattern, can be estimated inversely fromavailable observations. In practice, an intermediate vari-able G is often first calculated based on the followingrelationship:

C (𝜏) = G (𝜏)C (0)

where C(0)= x(0)x(0)′ and C(𝜏)= x(𝜏)x(0)′, which rep-resent the variance and lagged-covariance matrices of thestate vector, respectively.

In our case, the lag time 𝜏 is chosen to be 1 month.Once G(𝜏) is calculated, we can calculate B based on thefollowing formula:

B = 1∕𝜏∗ log (G (𝜏))

We can further calculate the noise statistics based onthe fluctuation dissipation theorem (Newman, 2007). Fora deterministic forecast, the equation is written as:

x (𝜏) = G (𝜏) ∗ x (0)

This is also the most likely state for a probability forecast(Penland, 1989).

Most previous LIM studies apply year-round data toestimate the matrix B and use the same B for all calendarmonths (e.g. Newman et al., 2013). An alternative versionof LIM is to estimate and apply a unique matrix B for

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Figure 7. (a) PC1 of tropical Pacific SST for August, November and their tendency and (b). Retrospective forecast comparison between observedSPPidx and predicted SPPidx by PC tendency model. Dotted lines represent the mean and standard deviation of SPPidx.


S. WU et al.

(a)

(b)

(c)

Figure 8. Temporal correlation between (a) Niño3.4 index, (b) PC tendency index, and (c) SPPidx with 500-hPa relative humidity for February.Significant values at 95% confidence level are shade. Thin solid lines represent positive values. Thin dashed lines represent negative values. [Colour

figure can be viewed at wileyonlinelibrary.com].

each of 12 calendar months (e.g. Xue et al., 1994, 2000),thereby allowing matrix B to be monthly dependent. Thisis particularly relevant in our case, because southern Peruprecipitation has very clear seasonality, suggestive ofdifferent mechanisms operating during individual months.

In practice, there are also too many zeroes for none rainymonths, which lead to unrepresentative variance matrixused to estimate LIM coefficients. Therefore, to build ourLIM model, we calculated the evolution matrix B for eachmonth.




DecExplained variance = 50.11% Explained variance = 48.20% Explained variance = 58.99% Explained variance = 47.26%

Explained variance = 13.95% Explained variance = 18.12% Explained variance = 15.42% Explained variance = 16.70%

7000

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Figure 9. First two EOF modes of southern Peru precipitation for December through March. The size of dots is proportional to EOF loadings.Red/dark and green/light colours represent different signs of EOF loadings. Solid red lines represent international and departmental boundaries.

Dashed blue lines represent the coast lines of ocean and lakes. [Colour figure can be viewed at wileyonlinelibrary.com].

EOF1

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7.28% variance explained 7.06% variance explained


64.7% variance explained 53.29% variance explained






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Figure 10. First three EOF modes of tropical Pacific SST for December through March. Thick solid lines represent “0”. Thin solid lines representpositive values. Thin dashed lines represent negative values. [Colour figure can be viewed at wileyonlinelibrary.com].

Because tropical Pacific SSTs and 200-hPa zonal windsare closely related to southern Peru precipitation, wechoose these two variables together with southern Peruprecipitation as state variables for our LIM model. In prac-tice, we use global geopotential heights (HGT200) insteadof 200 hPa winds, because geopotential heights are lessnoisy. To further reduce unwanted noise, we use PCs asso-ciated with the EOF modes of these variables, follow-ing Newman et al. (2009), to construct the LIM model.The first two modes of southern Peru precipitation, thefirst three modes of tropical Pacific SST, and the first twomodes of tropical-to-mid-latitude (0∘–360∘, 60∘S–60∘N)200-hPa geopotential height are selected (Figures 9–11).Adding the number of selected modes does not lead to sig-nificant improvement.

The first EOF mode (Figure 9) for southern Peru precip-itation shows a consistent monopole pattern for all monthsfrom December to March. The explained variance rangesfrom 47% in March to 59% in February. The second modeillustrates a dipole with opposing signs of the precipita-tion anomalies on either side of the Andes. The explained

variance ranges from 14% in December to 18% in Febru-ary. The station, Isla Taquile, which is located on an islandin Lake Titicaca, shows an exceptionally high loadingbecause its total annual precipitation and variability aremuch larger than at other stations, while it is noted thatremoving this outlier station does not result in any appre-ciable change in the EOF pattern. The time series associ-ated with the first EOF mode is dominated by inter-annualvariability, while the time series associated with the secondEOF mode is dominated by decadal variability (figures notshown).

The first EOF mode of tropical Pacific SSTs (Figure 10)is characterized by a typical ENSO-related horseshoe pat-tern with large SST variations in the central and easterntropical Pacific. The explained variance reaches a maxi-mum in December and minimum in March. The centre ofSST variations also shrinks from a large area across thecentral and eastern Pacific in December to a small area ofthe central western Pacific in March; correspondingly, theamplitudes also decrease dramatically. The second moderesembles the tropical Pacific meridional mode (TRMM)




S. WU et al.

Dec

Jan

Feb

Mar

Figure 11. First two EOF modes of global 200-hPa geopotential height for December through March. Thick solid lines represent “0”. Thin solidlines represent positive values. Thin dashed lines represent negative values. [Colour figure can be viewed at wileyonlinelibrary.com].

(Chiang and Vimont, 2004; Wu et al., 2010), in whichequatorial eastern Pacific SST anomalies have the oppo-site sign of those in the northeast tropical Pacific. The thirdmode resembles the nonlinear portion of the ENSO mode(Monahan and Dai, 2004).

The first two EOF modes of HGT200 (Figure 11) showpronounced variability across different months. In general,these spatial patterns do not persist as long as those ofSST and precipitation. Centres of action in HGT200 EOFmodes are mainly found across mid–high latitude regions.The first EOF mode essentially represents a contrastingpattern between the northern mid–high latitude regionsand the remaining portion of the globe, while the secondmode resembles the atmospheric response to the ENSOcycle. The explained variance ranges from 11 to 21%,which is notably smaller than that of precipitation and SST.

Nevertheless, the PCs related to these EOF modes formthe state vector of our LIM model:

x =⎧⎪⎨⎪⎩

PCs_PR

PCs_SST

PCs_HGT200

Once these PCs are predicted, the original field can bereconstructed based on the multiplication of the PC spatialpatterns and EOF time series. In order to be stable, theLIM model requires that all eigenvalues of the matrix Bbe negative. We have confirmed that our model meets thisrequirement.

In practice, our LIM model is initialized at the end ofDecember, with forecasts of subsequent precipitation inJanuary, February, and March carried out recursively. TheLIM and PC tendency models exhibit comparable forecastskill, with temporal correlations between predicted andobserved SPPidx of 0.68 and 0.66, respectively (Figures 7and 12). For individual stations and months, LIM per-forms much better than any single-variable regressionmodel (Figure 13), especially for stations with extremelylow climatological precipitation. For instance, the fore-cast skill for Tacalaya (station number 3 in Figure 1), asmeasured by the temporal correlation between predictedand observed precipitation, is 0.15 for the Niño3.4 modeland slightly higher, 0.23, for the PC tendency model. Incontrast, the LIM model has a substantially higher skillof 0.51 for that station’s precipitation prediction, whichrepresents an increase in accuracy of over 200%. Althoughthe forecast skill still has significant range among stations(Table 4), on average, the PC tendency model is superiorto the Niño3.4 model, and LIM model is further superiorto the PC tendency model.

Although LIM assumes linear dynamics, with the inclu-sion of random noise to represent nonlinear components,it fits very well to the statistics of many climate systems,such as tropical (e.g. Penland and Magorian, 1993) andmid-latitude climate system (e.g. Alexander et al., 2008).It also well represents the interactive processes betweenthe mid-latitudes and Tropics (Newman, 2007). One rea-son that the LIM model outperforms other simpler models




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Figure 12. Retrospective forecast comparison between observed (solid black) and predicted (dashed black) SPPidx (mm) by LIM model during1966–2016. Dotted lines represent the mean and standard deviation of SPPidx.

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Figure 13. Temporal correlation forecast skill by individual stations and month (during January, February, and March) of different predictive models.

is that the eigenvectors of evolution matrix B need not beorthogonal. This flexibility allows for interactions amongdifferent modes, leading to optimal growth of the initialanomalies (e.g. Penland, 1989; Alexander et al., 2008).

6. Conclusion and discussions

Precipitation variability over the Peruvian Andes has beenstudied widely over the past several decades, leading to animproved understanding of the primary responsible mech-anisms; however, there is little evidence that this has led todirect improvement in forecast skill of seasonal precipita-tion. A notable gap still exists between our understandingof the mechanisms and the development and operationaluse of forecasts.

In this article, we first re-examined the characteristicsof southern Peru precipitation and its relationship withENSO by using locally collected rain gauge data andreanalysis data sets. Then, we reviewed the current forecastskill of existing dynamical and statistical models. Lastly,we introduced two simple statistical models to predict

Table 4. Mean temporal correlation forecast skill by month dur-ing January–March, based on individual stations’ precipitation,according to the Niño3.4 regression model, SST PC tendency

model, and LIM model.

Months Niño3.4 model PC tendency model LIM

Jan 0.20 [0.01–0.42] 0.27 [0.09–0.51] 0.38 [0.14–0.62]Feb 0.41 [0.23–0.56] 0.49 [0.32–0.65] 0.55 [0.33–0.69]Mar 0.18 [0.00–0.30] 0.27 [0.07–0.45] 0.25 [0.05–0.40]

Numbers in brackets indicate the range of skill (measured by temporalcorrelation) across the 29 stations.

southern Peru precipitation anomalies at both the regionalscale and for individual stations, with both models demon-strating improvement over existing models based on retro-spective forecast experiments.

One common critique for statistical modellingapproaches is the likelihood of overfitting based onhistorical data. This may have some effect on our PCtendency model as we performed predictor screeningusing the same data we validate on, thus even thoughwe carried out cross-validation, the skill may be slightly


S. WU et al.

Table 5. 2017 real-time forecast verification.

Climatological mean SPPidx (CLM) 312 mm

Wet events (CLM+1*standard deviation) 405 mm2017 station observations 392 mm (+)Niño1+2 forecast 288 mm (−)Niño3.4 forecast 336 mm (+)PC tendency model forecast 348 mm (+)LIM forecast (TRMM data as DEC initialcondition)

390 mm (+)

‘+’ represents wet, and ‘−’ represents dry conditions.

overstated as pointed out by DelSole and Shukla (2009).The LIM model, however, does not suffer from thisproblem, because no pre-screening is carried out.

To address this, we verified the models using the new2017 data, by issuing forecasts in December 2016. Itshould be noted that the real-time HADISST data andobserved precipitation station data are typically not yetavailable at the end of 2016; therefore, we used real-timeNOAA Optimal Interpolation SST (OISST) and TropicalRainfall Measuring Mission (TRMM) data (Raynoldset al., 2007; Banzon et al., 2016) as alternatives. Wepredicted that JFM 2017 would be slightly wetter thannormal (Table 5), which turned out to be accurate based onobservations. These results also provide further evidencethat the LIM and PC tendency models are superior to thesimple Niño indices models. We acknowledge that 2017is only a single case, and additional validation years aredefinitely warranted.

Multi-model integration of all available model fore-casts should be investigated. For example, our consensusdemonstrates that if the dynamical models predict anoma-lously wet conditions, then the reality tends to be wet eventhough dynamical models have an obvious dry bias. Thisis also the case for 2017 in which most dynamical modelsissued forecasts of wetter than average conditions. There-fore, for a wet scenario, we may be able to put more weightinto dynamical model predictions, while for dry years,more weight on the statistical models may be justified.

Acknowledgements

This study was supported by Southern Peru Copper Cor-poration. Rain gauge data are provided by SENAMHI andSPCC. NCEP–NCAR Reanalysis data were provided bythe NOAA/OAR/ESRL PSD, Boulder, CO, USA, fromtheir website at http://www.esrl.noaa.gov/psd/. We thankCarlos Sanchez for insightful discussions during the prepa-ration of the work. We also thank the anonymous reviewersfor their careful reading of our manuscript and their manyinsightful comments and suggestions.

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https://doi.org/10.1175/1520-0442(2004)017,3026:TSATSO.2.0.CO;2

https://doi.org/10.1175/1520-0442(2004)017,3026:TSATSO.2.0.CO;2

https://doi.org/10.1029/2002JD002670

https://doi.org/10.1029/2002JD002670

https://doi.org/10.1007/s00704-015-1421-2

https://doi.org/10.1002/joc.5185

https://doi.org/10.1002/joc.5185

https://doi.org/10.1038/ncomms8154

https://doi.org/10.1038/ncomms8154

https://doi.org/10.1175/JCLI-D-12-00113.1

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