A Mesoscale Data Analysis and Downscaling …...A Mesoscale Data Analysis and Downscaling Method over Complex Terrain REINHOLD STEINACKER,MATTHIAS RATHEISER,BENEDIKT BICA,BARBARA CHIMANI,MANFRED

A Mesoscale Data Analysis and Downscaling Method over Complex Terrain

REINHOLD STEINACKER, MATTHIAS RATHEISER, BENEDIKT BICA, BARBARA CHIMANI, MANFRED

DORNINGER, WOLFGANG GEPP, CHRISTOPH LOTTERANER, STEFAN SCHNEIDER, AND SIMON TSCHANNETT

Department of Meteorology and Geophysics, University of Vienna, Vienna, Austria

(Manuscript received 15 June 2005, in final form 30 November 2005)

ABSTRACT

A mesoscale data analysis method for meteorological station reports is presented. Irregularly distributedmeasured values are combined with measurement-independent a priori information about the modificationof analysis fields due to topographic forcing. As a physical constraint to a thin-plate spline interpolation, theso-called “fingerprint method” recognizes patterns of topographic impact in the data and allows for thetransfer of information to data-sparse areas. The results of the method are small-scale interpolation fieldson a regular grid including topographically induced patterns that are not resolved by the station network.Presently, the fingerprint method is designed for the analysis of scalar meteorological variables like reducedpressure or air temperature. The principles for the fingerprint technique are based on idealized influencefields. They are calculated for thermal and dynamic surface forcing. For the former, the effects of reducedair volumes in valleys, the elevated heat sources, and the stability of the valley atmosphere are taken intoaccount. The increase of temperature under ideal conditions in comparison to flat terrain is determined ona 1-km grid using height and surface geometry information. For the latter, a perturbation of an originallyconstant cross-Alpine temperature gradient is calculated by a topographical weighting. As a result, thegradient is steep where the mountain range is high and steep. If, during the interpolation process, somesignal of the idealized patterns is found in the station data, it is used to downscale the analysis. It is shownby a cross validation of a case study that the interpolation of a mean sea level pressure field over the Alpineregion is improved objectively by the method. Thermally induced mesoscale patterns are visible in theinterpolated pressure field.

1. Introduction

Drawing the most accurate surface analyses was theambition of meteorologists some 30 yr ago. Interpolat-ing scattered station reports by hand required experi-ence and knowledge about the physics that govern thesurface fields. Today, computers generally do this job,but the requirements remain the same. So, the workpresented in this paper is about a mesoscale data analy-sis method that imitates an experienced meteorologistby applying some knowledge about the physical pro-cesses involved.

Meteorological fields over complex terrain are highlyinfluenced by the earth’s surface in various ways de-pending on the properties of the surface and on thescale of interest. Interpolation of irregularly spaced sta-

tion data to a regular grid becomes challenging if to-pography is complex and the fields have to be repre-sentative for the lower meso-� or even upper meso-�scale. In this case, typical length scales range from 5 to50 km for which grid resolutions from 2 to 20 km areneeded. Because the resolution of a synoptic stationnetwork is usually lower, the interpolation process re-quires downscaling of information. The aim of thiswork is to account for additional physical knowledgeabout the distribution of the variables between the sta-tion locations. Otherwise, the interpolation under thecircumstances mentioned above might not be physicallyappropriate (though mathematically correct) becausethe influence of the complex topography would not beadequately taken into account. An important premisefor the current work is that the proposed algorithmcreates useful results without a first-guess field of anumerical weather prediction model or climatologicalbackground information. There are two reasons for thisstep: First, the transparent and direct approach forsimulating the thermodynamic processes over complex

Corresponding author address: Dr. Reinhold Steinacker, Dept.of Meteorology and Geophysics, University of Vienna, Alt-hanstraße 14 1090, Vienna, Austria.E-mail: [email protected]

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terrain ensures the understanding of the physical pro-cesses involved. Second, the method remains objectiveand independent when it is used for validating forecastfields from numerical prediction models.

There exist many interpolation techniques that arebased on mathematical or statistical concepts. Early al-gorithms used least squares polynomial fitting (e.g.,Panofsky 1949) or inverse distance fitting (e.g., Cress-man 1959) for the calculation of values between thesampling points. The purpose of these methods was tocalculate an objective analysis (in contrast to the sub-jective hand analyses) as an initial field for numericalexperiments. They were used for synoptic-scale analy-ses in the free atmosphere (e.g., at 500 hPa) wheresurface influence is negligible. With the work ofBergthorsson and Döös (1955) and Thompson (1961),the concept of background or first-guess fields was in-troduced to objective analysis. They were calculatedfrom a short-term numerical model or climatologicalinformation. One popular method, the so-called succes-sive correction scheme, was introduced by Barnes(1964). Starting with a first-guess field, several succes-sive iterations of corrections are being applied to thegridded fields.

The statistical methods represent another type of ob-jective analysis. The so-called optimum interpolationmethod (e.g., Gandin 1965) is a minimum variance al-gorithm that utilizes statistical information gained froma large dataset. Multivariate statistical methods allowimposing physical constraints between two or morevariables that are interpolated at the same time (e.g.,Lorenc 1981). Anisotropic structures of the fields,forced by complex topography, are not easy to incor-porate by this approach (Lanzinger and Steinacker1990).

A special type of functional fitting method is con-strained minimization. This interpolation approach al-lows dynamical constraints to be imposed (e.g., Sasaki1958, 1970). A special case of the constrained minimi-zation is the spline method (Reinsch 1967). Besidesminimizing the squared differences between measure-ments and interpolated values (cost function) smooth-ing constraints may be applied (penalty function). Ifthese mathematical constraints are chosen in a physicalcontext, they are implicitly also physical constraints.

The use of a spline algorithm as a basis for the fin-gerprint technique, introduced in this paper is suggestedfor two main reasons: First, the above-mentioned physi-cal constraints may be imposed mathematically in anelegant way. Second, the inclusion of additional a prioriknowledge represents a powerful application of down-scaling. The term “fingerprint” was first used in clima-

tology by Hasselmann (1993), as a statistical method forthe detection of climate change signals in observationsor climate models (see the discussion in Levine andBerliner 1999).

The term “downscaling” is mainly used for imposingregional scale patterns on global scale general circula-tion models (GCMs). There are statistical methods(e.g., von Storch et al. 1993) and dynamical methodsthat combine a GCM with small-scale models (e.g.,Zagar and Rakovec 1999). There has also been workthat focuses on downscaling observational data. For ex-ample Frei and Schär (2001) and Schmidli et al. (2001)use empirical orthogonal functions (EOFs) to recon-struct precipitation fields over the Alps for a period of15 yr. With the EOFs, most dominant patterns of theprecipitation field can be found objectively.

Increased computer power and the installation ofmesonets have led to the development of new high-resolution mesoscale analysis systems. Lazarus et al.(2002), provides an overview of several of U.S. analysissystems. They use classical interpolation methods likeoptimum interpolation for assimilating data from vari-ous sources and are often coupled with numerical mod-els.

To take into account the spatial anisotropy of analy-ses over complex terrain, two approaches have recentlybeen presented. Deng and Stull (2005) suggest a modi-fication of the background error correlation matrix bytopography. This leads to a reduction of influenceacross mountain ranges and hence may sharpen gradi-ents. A second approach was suggested by Myrick et al.(2005) with their “mother–daughter” method. Depend-ing on the elevation difference between neighboringgrid points the information is anisotropically distrib-uted across the domain.

Little can be found in the recent literature dealingwith the utilization of physical knowledge about topo-graphical, land use, or other surface forcing of the at-mospheric boundary layer for downscaling atmosphericfields beyond the resolution of observational data. Inprinciple, a background field of a high-resolution nu-merical model could be utilized, to refine the analysis indata-sparse areas in a physically consistent way. Theresolution of today’s operational mesoscale predictionmodels, however, is still not sufficient to resolve thescale of Alpine valleys, which has a strong impact onobservational data. Furthermore, the use of model datalimits the use of the analysis for validating the modelfields. Thus, the method described in this paper, thefingerprint technique, is a new, numerical forecastmodel-independent way of downscaling irregularly dis-tributed observational data. The basics of this modi-

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fied-spline interpolation scheme and its implemen-tation are explained in section 2. In our context, thefingerprints are idealized predefined patterns, repre-senting physical forcing mechanisms by topography, be-ing superimposed on the field of surface measurements.The physical concept behind two selected fingerprints isdescribed in section 3. In section 4, a case study is pre-sented showing improvement of the mean sea levelpressure field analysis over the Alps by objective mea-sures.

2. Outline of the fingerprint method andimplementation in an analysis scheme

With the aid of a one-dimensional example the math-ematical concept of the fingerprints is discussed in thefollowing.

Figure 1 shows a one-dimensional domain with 13grid points along the x axis. The scalar variable �(x) isgiven at three locations by measurements (crosses).Here �(x) is influenced by a topographic forcing on ascale, which is not resolved by observations. Thus, acertain portion of the measured values can be assignedto the mountain influence. If the latter is separatedfrom the contributions of other (e.g., large scale) forc-ing, a weighted contribution (squares in Fig. 1) of theso-called idealized fingerprint �m(x) (dotted line in Fig.1) remains. Because �m(x) is known at every grid point,knowledge about the behavior of �(x) is also trans-ferred (i.e., downscaled) from the station locations toevery grid point. The fraction of the observed value thatremains after subtracting the weighted �m(x) from�(x) (triangles minus squares) is the part [�S(x); dia-monds] that is unexplained by the topographic forcing.Hence, at each grid point the value of the scalar quan-tity may be formally split into two parts:

��x� � �S�x� � cm�x��m�x�. �1�

The coefficient cm(x) is a variable weighting factor.Here, �S(x) and cm(x) are not known a priori. How-ever, �m(x) is a precalculated idealized field, based onthe high-resolution elevation data (see section 3). Thescaling factor cm(x) varies in space and from case tocase. This is necessary because �m(x) represents valuescalculated for ideal conditions, but the actual influenceof the topography on �(x) varies, because of a diurnal,seasonal, and weather-type dependence.

The idea of including additional information to mea-surement values goes together with the interpolation ofirregularly distributed, sparse data to a regular grid.The method used is a constrained least squares fitting

method, a spline method, first described by Reinsch(1967):

I � �k�1

3

wk��xk� �O�xk��2 � �J��, with

J�� 1

13

��2�

�x2�2

dx. �2�

Equation (2) is related to Fig. 1, and k indicates thenumber of observations. For the analysis, the weight wk

is set to a very high value, so that the difference be-tween the analyzed value � and the observed value �O

becomes 0. In practice, this is equivalent to reducing thecost function I to J only. This implies that the interpo-lation function exactly matches the observed values.Hence, it is important that an error detection and cor-rection procedure for systematic and gross errors is car-ried out in advance (e.g., Steinacker et al. 2000). Thepenalty function with second-order derivatives is usedas a constraint for the smoothness of the interpolatedfield, which is equivalent to the cubic-spline algorithm.The observed values are interpolated to the grid pointstogether with the incorporation of the topographic in-formation. This is achieved by substituting Eq. (1) intoEq. (2) and means that the smoothness condition is onlyapplied to that part of the field �S(x), which is unex-plained by the topographic pattern:

I � �1

13

��2�S�x�

�x2 �2

dx

� �1

13

� �2

�x2 ��x� cm�x��m�x��2

→ Min. �3�

Equation (3) shows the variational approach mini-mizing the cost function I, which consists of the known�m(x) and the (except at data points) unknown values�(x) and cm(x). The expression cm(x)�m(x) adjoins theinfluence of the topography into the interpolation. Tosolve the cost function, the discretized version of Eq.(3) is used:

�i�2

12 ��i�1 2�i � �i1

��x�2

cm,i��m,i�1 2�m,i � �m,i1

��x�2 ��2

→ Min. �4�

Now the cost function has to be derived with respectto all unknown grid values �i and the weighting factors


cm,i and then set equal to 0. In general, the resultingequation system is well defined and may be solved inorder to obtain the interpolated values and cm,i. Theresult is an interpolation field (triangles in Fig. 1) thatfeatures the topography-induced pattern superimposed

to the smooth unexplained part on a larger scale. Thedifference between this and a spline interpolation(circles in Fig. 1) without taking into account any addi-tional information becomes evident.

The cm,i will only be 0, if there is no signal of the

FIG. 1. Schematic 1D domain showing the interpolation of an arbitrary scalar variable, given at irregularlydistributed points, to a regular grid. (top) Measurements are indicated by crosses and triangles denote the finalinterpolation �(x) consisting of a part unexplained by the fingerprint (diamonds; �S) and the weighted fingerprint(squares; cm�m). (bottom) Cubic-spline interpolation without taking into account topographic information (dia-monds) and the idealized fingerprint (squares; �m).


influence of the topography in the observations at all.The better the correlation between the measurementsand the fingerprint field is, the smoother the unex-plained part will be. If the correlation is negative, thecm,i will also be negative (i.e., the topographically in-duced pattern is reversed). Note that in general the cm,i

must be allowed to vary within the domain. Then afurther constraint (e.g., a minimum gradient or mini-mum curvature) of the weighting factors has to be in-troduced to the cost function. If there are only threedata points available, like in the schematic example ofFig. 1, just one value for cm,i for the whole domain canbe set.

The fingerprint method improves the resolution of anexisting topographic signal. In Fig. 1 it is well repre-sented by the three available measurements, becausetheir variability corresponds nicely to the variability ofthe fingerprint signal. Note that the interpolation stillleads exactly through the given values as a result ofsetting � �O � 0 at data points in the cost function.

3. The calculation of the idealized topographicfingerprints

The fingerprint method uses measurement-independent information on the behavior of interpo-lated quantities in addition to station values. This sec-tion will describe how this extra information is obtainedand quantified for the Alpine region in central Europe.

a. The thermal fingerprint

The example shown in Fig. 1 may be interpreted as athermal influence of the topography on the field of re-duced pressure. Formulating the fingerprint method fortwo dimensions, the thermal influence of mountainousregions may be quantified as idealized fields [�m in Eq.(1)] by a strictly physical approach. The effect is quan-tified in terms of the idealized thermal fingerprint. Thebasic idea is to calculate the spatial temperature distri-bution for idealized radiative heating or cooling over

mountainous terrain and to use the hydrostatic relationfor estimating the corresponding pressure field. If un-disturbed daytime conditions are assumed, both, thereduced air volume over topography and a free volumeover a plain will experience the same radiative flux atthe top of the volumes. If the surface albedo, the sen-sible and latent heat fluxes, and the ground heat fluxare the same over mountainous terrain and the plains,and the energy gain or loss is evenly distributed withinthe volume, the ratio of the overall heating or coolingwill be given exactly by the ratio of air volumes. Thedistribution of energy gain or loss, however, will hardlyever be evenly distributed but strongly depending onthe static stability and the area–height distribution.Thus, the excess of the diurnal amplitudes of tempera-ture over mountains as compared with the plains can-not be given by a constant ratio. Steinacker (1984) de-scribes this effect for stably stratified air thermallyseparated layer by layer in valleys. The higher the sta-bility and the more convex the area–height distributionof a valley (see Fig. 2), the larger the resulting ratio ofheating or cooling between the valley and the plain.The thermal separation of stably stratified layers in val-leys is evident from observations. The temperature dif-ference between the slope layer, where the diabaticheating or cooling takes place, and the “free” valleyatmosphere stays roughly constant during the daytime(time with upslope winds) or nighttime (time with downslope winds). This means that the same heating/coolingamount that the slope layer experiences is immediatelytransferred to the free atmosphere layer by layer byadiabatic heating/cooling. This concept has been intro-duced by Vergeiner and Dreiseitl (1987).

The topographic amplification factor (TAF), a meanratio of the diurnal amplitude of temperature in thevalley to that at the same altitude over the plain, ne-glecting the static stability variation of the air, has beenintroduced by Whiteman (1990). The TAF may be for-mulated by using the volume reduction of the air withinthe valley solely due to its geometry.

FIG. 2. Prototypes of area–height (A–z) distributions (hypsographic profiles): (left) concave,(middle) constant gradient, and (right) convex. The area within a given domain where theterrain lies above height z is represented by A.


In Eq. (5) V is the total volume of the valley (solidmatter and free air) and Vred is the volume that is leftafter subtracting the space filled with solid matter:

TAF �V

Vred�

��x�2�zcrest zvalley floor�

��x�2�zcrest zmean�. �5�

Instead of using the overall TAF of Eq. (5), a differ-ential approach is applied. The Alps are divided into athree-dimensional grid (grid size �x � �y � 10 km and�z � 500 m). Horizontally the domain (see, e.g., Fig. 7)spans over 1580 km 1160 km with the southwestern-most grid point at 41.45°N, 3.47°E, and vertically thereare nine layers. Elevation data is available on a 1-kmgrid, thus there are 100 values available in every10 km 10 km grid cell.

For each layer h of every grid box m, the mean alti-tude zm,h is calculated as the arithmetic mean of all 100height points which lie within the respective 10 km 10 km grid cell, after having modified their height to theheight of the lower/upper level, if the grid point isabove/below this level. The arrows in Fig. 3 symbolizethe modification for the calculation of zm,1. In the one-dimensional schematic of Fig. 3 zm,1 is given by themean of the six height values that range between 0 and500 m plus four height values of 500 m; zm,2 by the twovalues between 500 and 1000 m, six height values of 500m, and two height values of 1000 m; and finally zm,3 bythe two values between 1000 and 1500 m and eightheight values of 1000 m. A volume element is assumedas filled by solid matter up to zm,h (shaded area in Fig.3). The energy input/loss by sensible heat flux for eachvolume depends on the surface area (slope area pro-jected horizontally) within each layer. In our cross-sectionexample in Fig. 3 six topography grid points are be-tween 0 and 500 m and two are between 500 and 1000 m

and 1000 and 1500 m, respectively, hence A1/A � 0.6,A2/A � 0.2, and A3/A � 0.2 (see Fig. 4).

The ratio of the volumes of a complete air parcel tothat of a partially filled one multiplied by the ratio ofthe differential area to the total area can be seen as adifferential mass specific TAF, expressed by pressurevalues p from different heights:

DTAFM�h� �p�zlayer bottom� p�zlayer top��h

p�zmean� p�zlayer top��h

Ah

A. �6�

The differential topographic amplification factor(DTAF) of layer 1 in Fig. 3 (DTAFM-1) is larger than1, the DTAFM-2 is less than 1 and the DTAFM-3 is evensmaller (see Fig. 5). The DTAFM of layer 4 and aboveis 0, because the highest elevation of the 10-km grid celllies below 1500 m. The DTAFM of a grid cell over flatland at sea level is 1 for the lowest layer and 0 for allhigher layers.

The air of every element has an initial value for tem-perature and pressure taken from the U.S. StandardAtmosphere, 1976 [(U.S. Committee on Extension tothe Standard Atmosphere) COESA 1976]. Now the in-crease of temperature depending on the DTAF is cal-culated for all the nine layers in every column. First thetemperature increase is computed for a free volume(without solid matter) by

��T�h �SH Ah

cp Mh�t

�g SH Ah �t

cp Ap�zmean� p�zlayer top��h

�g SH �t

cpp�zlayer bottom� p�zlayer top��hDTAFM�h�.

�7�

FIG. 3. Schematic cross section through a 10-km grid cell partitioned into the four lowest500-m layers. The dots represent the topographic height on the 1-km grid. The zm,h (grayshading) are the mean heights filled by solid matter within each layer.


Taking the pressure at the bottom p(zlayer bottom), atthe top p(zlayer top), and at the mean height p(zmean) ofthe layers from the standard atmosphere only values forthe sensible heat flux SH and the duration �t have to bechosen to compute the temperature change within thelayer with Eq. (7). Here g is the acceleration of theearth and cp the specific heat at constant pressure. Aswe only need the fingerprint pattern and not its ampli-tude, arbitrary values for the sensible heat flux and theheating duration were chosen: SH � 150 Wm2 and�t � 6 h.

The result is a temperature profile determined for alllayers (Fig. 6). Because superadiabatic lapse rates arenot realistic, convection is simulated in this case: thesuperadiabatic rate is reduced by transferring the ex-cess energy to higher layers. Of course this “convec-tion” is weighted with respect to the mass of each af-fected layer. For layers that are completely beneath the

surface, the standard lapse rate is used to create a fic-titious temperature profile down to sea level. This cor-responds to the standard procedure for pressure reduc-tion. Now the corrected vertical temperature profile isturned into a pressure profile: the hydrostatic equationis applied from layer to layer starting at the top of thecolumn of air at 4500 m with the standard atmospherepressure. At each 500-m level below 4500 m down tothe sea level, a pressure value is calculated hydrostati-cally from which the corresponding standard pressure issubtracted. This difference at sea level represents thethermal fingerprint value.

The resulting thermal fingerprint field has a 10-kmresolution (Fig. 7), because the grid points used for

FIG. 5. DTAFM of the four layers in Fig. 3 (boldface line) andof a plane grid cell (boldface dotted line).

FIG. 6. Computed vertical profile of temperature before (stan-dard atmosphere �) and after radiative heating (dashed with graydots) in the four lowest layers of an arbitrary air column. Here �symbolizes the dry adiabatic lapse rate. The superadiabatic part inthe lowest level is mixed mass weighted with the upper levels(boldface solid with black dots).

FIG. 4. Energy flux relevant areas for each layer: Ah is the fraction for the hth layer of thetotal area A of a 10-km grid cell. Values of Ah depend on the number of height points in thelayer h (cf. Fig. 3).


calculating the mean heights of the air parcels weretaken from a 10-km grid. However, the topography in-formation, given on a 1-km grid, represents a realisticvolume reduction because of the valley shapes and el-evated heat source distribution. So, the 1-km informa-tion is implicitly included in the fingerprint. The choiceof the size of the influence squares is a matter of thescale of valleys one wants to resolve. The volume re-duction is best represented if the grid size equals themean valley width. Choosing 10-km squares means re-solving the typical Alpine valley.

Note that the isolines roughly follow the shape of themountains but the field is not equivalent to a smoothedtopography. Besides the effect of an elevated heatsource, the volume effect is apparent, for example, inthe Adige valley (2 in Fig. 7), the Rhone valley east ofLake Geneva (1 in Fig. 7) or the Aosta Valley (3 in Fig.7). Those major Alpine valleys have a nearly horizontalvalley floor but nevertheless show a significant along-valley pressure gradient. Also, the pressure gradient intributaries of the large Alpine valleys, where both, vol-ume and elevation effects play a role, is representedreasonably. The isolines in Fig. 7 represent perturba-tions of the pressure with respect to the plains at sealevel. The magnitude of the perturbation is not impor-

tant, because the idealized values are being scaled by aweighting factor when they are applied (see section 2for details).

b. The dynamical fingerprint

When an airflow impinges upon mountains, it ispartly deflected around, partly deflected over the ob-stacle, or even sometimes becomes blocked. This basi-cally depends on the scale of the mountain, the staticstability, and the speed of the air, a measure for this isthe Froude number (Chen and Smith 1987). Flow de-flection or blocking of the air needs a correspondingpressure perturbation. Typically, for mountains the sizeof the Alps, a positive pressure perturbation is seen onthe windward side (Stau) and a negative pressure per-turbation is seen on the lee side. These pressure per-turbations can be explained hydrostatically [i.e., on thewindward (leeward) side we see a negative (positive)temperature perturbation]. The creation of an idealizedtemperature perturbation field, which is then used as adynamical fingerprint, is carried out as follows: Theprocedure obeys the rule that the higher and steeperthe topography, the greater the eventual temperaturegradient across the mountain range. We start with aconstant gradient field (north–south and west–east) and

FIG. 7. Idealized thermal pressure fingerprint for the Alps with 10-km resolution. Isolines are sea level pressuredifferences to flat terrain (1-hPa interval). Gray shadings denote terrain height above sea level.


obtain the perturbation by an iterative modification ofthe three-dimensional field [T(i, j, k)] by a topographi-cal weighting:

Tnew�i, j, k� �

�l�i1

i�1

�m�j1

j�1

�n�k1

k�1

w�l, m, n�Told�l, m, n�

�l�i1

i�1

�m�j1

j�1

�n�k1

k�1

w�l, m, n�

.

�8�

The weights w(x, y, z) are the smaller, the higher theterrain above the grid point is. Three cases are beingdistinguished:

1) terrain height below gridpoint height (zt � zg): w(x,y, z) � 1,

2) terrain height at most 500 m above gridpoint height:w(x, y, z) � exp[(zg zt)/200 m], and

3) terrain height more than 500 m above gridpointheight: w(x, y, z) � exp (2.5).

The grid point values at the uppermost level (4500 m)are frozen, at the lowest level (500 m) and at thelateral boundaries the values are successively set equalto the next grid points within the domain.

If the terrain height is more than 500 m above thegridpoint height, according to case 3, the weight re-mains constant at around 10% of case 1. It has to befurther noted that the spatial anisotropy of meteoro-logical fields (i.e., the lower correlation of data in thevertical as compared with the horizontal) is taken intoaccount by using vertical weights only 1/1000 of thehorizontal. The grid spacing for computing the dynami-cal fingerprint is 2 km horizontally and 500 m vertically,covering the Alpine region and its surroundings.

Because of the high resolution and the high numberof grid points there are many iterations necessary untilthe maximum difference between two successive itera-tions becomes less than a given threshold value. In thelast step, the initial constant gradient field is subtractedfrom the modified field that yields to the temperatureperturbation. As with the thermal fingerprint, the com-putation of the dynamical pressure fingerprint is carriedout by hydrostatic integration from the uppermost level(4500 m), where no disturbance is being assumed, downto sea level. The dynamical pressure fingerprint at sealevel for a northerly flow is shown in Fig. 8. Like thethermal fingerprint, the field is plotted in 10-km reso-lution. The dipole structure is not found in nature, be-cause the northerly (undisturbed) flow requests a west-

FIG. 8. Idealized dynamical pressure fingerprint for the Alps with 10-km resolution. Isolines are sea levelpressure differences to flat terrain (1-hPa interval).


ward pointing pressure gradient. If such a pressure fieldis added, the result (Fig. 9) becomes quite similar towhat is commonly observed during north foehn flows(i.e., a “foehn nose” becomes well pronounced).

With a southerly flow, the configuration of the sign ofthe dynamical fingerprint reverses. Together with thewest–east fingerprint presently three different finger-prints are used for the operational analyses. For thispurpose, the right side of Eq. (3) is expanded towardthree weights multiplied with the three predefined fin-gerprint values.

4. Case study showing the fingerprint effect

This example documents how the accuracy of theinterpolation of a thermal high over the Alps is im-proved by the thermal fingerprint. Prior to the interpo-lations the gross and systematic errors were corrected.A spatial self-consistency check was applied (see Stein-acker et al. 2000).

Figure 10 shows the interpolated field of mean sealevel pressure on 0300 UTC 31 March 2002 (approxi-mately 11⁄2 hours before sunrise). The interpolation wasdone simply by cubic-spline interpolation (withoutdownscaling by fingerprints). The isopleths are drawn

in an interval of 1 hPa. The crosses are stations withpressure measurements used for the interpolation. Thenight was clear with weak winds in most of the Alpineregion. There is a weak pressure gradient with a ther-mal high showing two centers, one in the eastern Alps,and one in the western Alps. The highest pressure val-ues are about 1025 hPa. The interpolation using thethermal fingerprint (i.e., downscaling) results in a moredetailed pressure field (Fig. 11). The patterns over theplains have not changed, but over the mountains thecenters of the high are more pronounced (up to 1027hPa) in shape and intensity, especially over the westernAlps where the station density is very low. The valuesof the existing stations contain the mountain inducedpart of the pressure field �m(x) from Eq. (3). The situ-ation is comparable to the schematic one-dimensionalsetup in Fig. 1. In Table 1 the pressure at the stations inthe valleys (annotated with 1, 5, and 9) is higher than atthe foothills. The lack of detail in the analysis becauseof the low station density in large areas of the westernAlps is compensated by the physical a priori knowledgeof the thermal fingerprint: Fig. 11 shows more details inthe analysis by enhancing the size, shape, and intensityof the thermal high.

In the eastern Alps the station density is higher than

FIG. 9. Idealized dynamical pressure fingerprint combined with westward-pointing pressure gradient. Contourlines are every 1 hPa.


FIG. 10. Analysis of MSL pressure without fingerprint technique (1-hPa intervals); crosses represent stationreports used for the analysis.

FIG. 11. Same as in Fig. 10, but with the fingerprint technique. Stations with numbers and point A areexplained in the text. Stations 6 and 7 are just outside the domain.


in the western Alps. Note that the area of the thermalhigh is well equipped with stations. Therefore its inten-sity and shape is quite well resolved by the measure-ments. In this case only the details are improved, like itsextension toward the northeast.

Around the three-corner point of Germany, Austria,and the Czech Republic (point A in Fig. 11), the ther-mal high is very well improved by the fingerprint tech-nique. The situation over the Balkan Mountains, wherethe station density is again very low, is similar to thewestern Alps: there is a weak signal of a thermal highbetween stations 10 and 11, which is interpreted by thefingerprints. Another good example is the Adige valley.

The lower pressure with respect to the adjacent areas tothe left and to the right is represented by station 12.With the fingerprint, however, the shape and positionof the mesoscale trough are improved. The same ap-plies for the valley with station 10 in the Balkan Moun-tains and other major valleys.

Figure 11 shows the field that is expressed by �(x) inFig. 1. Hence, this is the quantity that contains both thepart �S(x) that is unexplained by topography and thetopographic-induced part of the measurementcm�m(x). As explained above �m is known and � andcm are calculated while solving the minimum the crite-rion in Eq. (4). See Fig. 12 for the field of the topog-raphy-induced part. As explained above, the highestvalues of topographic influence (up to �5 hPa) in themeasured values are in the western Alps and over theBalkan Mountains. Over the plains this part is 0. Figure13 shows the unexplained part. The distribution of�S(x) is very smooth, because it does not contain all thesmall-scale influences induced by the mountains. Thisphysical splitting of each analyzed field allows new andexciting evaluation approaches.

To prove the fact that the use of the fingerprint tech-nique objectively improves the interpolation, a crossvalidation was carried out. For the analyses in Figs. 10and 11 data from a set of 260 meteorological stations

FIG. 12. Field of orographic influence [cm�m in Eq. (1)]. Isolines with 1-hPa intervals.

TABLE 1. MSL pressure (MSLP) values (hPa) for selectedstations (cf. Fig. 11).

Station MSLP

1 1025.92 1021.33 1020.34 1021.35 1023.96 1022.47 1022.58 1023.49 1023.8


was used. One station datum after the other was left outand a value of the pressure field was interpolated to thelocation of the corresponding suppressed station for ev-ery of the 260 analyses. A simple inverse distance in-terpolation was done because in general the stations arenot located on grid points. The difference between themeasured station values and the interpolations (i.e., theinterpolation error) was statistically evaluated for thetwo fields (see Table 2).

In general, using the spline method yields larger in-terpolation errors than the fingerprint method. For ex-ample, the maximum difference between the measure-ments and the interpolated values occurs for the stationin Sarajevo (9.82 hPa). Mesoscale patterns in the MSLpressure field are not resolved by the few stations in

that area. Using the fingerprint technique reduces theinterpolation error to 2.09 hPa for Sarajevo. The statis-tics confirm this result: the rms and the standard devia-tion of the interpolation errors are about 25% lower ifthe fingerprints are included to the analysis. The rangeof the differences is reduced by about 75%.

Not all of the examined stations are situated in areaswith large topographic influence. Comparing Fig. 10with Fig. 11 shows that the improvements of the inter-polation by the fingerprints are caused by the stationswithin the Alps only. A statistics concerning the sta-tions in mountainous terrain only would show evenmore impressive improvements.

5. Conclusions and outlook

By combining physical knowledge about the behav-ior of the atmosphere over complex terrain with mea-surements from stations, the presented method createsanalyses featuring subscale, topographically inducedpatterns. Station data downscaling is done indepen-dently from numerical models by calculating idealizedinfluence fields from very high resolution topographicinformation. No first-guess or background fields arenecessary. The method works without statistical infor-mation like long time series.

There are many applications for this objective data

TABLE 2. Statistics of the differences between station measure-ments and interpolated values at station locations with units ofhectopascals and the number of meteorological stations n � 260.

Downscaling by fingerprints Spline interpolation

Max diff 2.57 9.82Min diff 1.22 4.86Range 3.79 14.68Mean value 0.13 0.25Std dev 0.34 0.43Rms diff 0.36 0.50

FIG. 13. Field without orographic influence (� cm�m). Isolines with 1-hPa intervals.


analysis method. Relatively low computational costs al-low real-time calculations with hourly data from theGlobal Telecommunications System. The two-dimensional surface maps of reduced pressure, horizon-tal wind, potential temperature, or equivalent potentialtemperature have proved to be very useful in nowcast-ing. They help detect front signals and predict theirpropagation. They also offer a quick overview of thecurrent weather situation and allow a detailed look atmesoscale surface patterns like vortices. Because themethod is model independent, the analysis fields areideal for model validation. Maps of the differences be-tween predicted and analyzed fields show model derail-ments at first glance. Both the analysis maps and themodel validation have been successfully used by opera-tional forecasters of the Austrian Aviation Service in areal-time operational mode for the past five years.

Other mesoscale climatology applications are in de-velopment. Long-term time series of gridpoint valuesallow for the creation of a climatology in a high tem-poral and spatial resolution.

The concept of fingerprints may be extended to otherforcings inducing small-scale surface features in objec-tive analyses. Thermal and dynamic processes are alreadydescribed in this paper. The influence of cities (heat islandeffect), lake or sea shores (land–sea-breeze systems) arephysically well understood. Knowing the appropriatesurface information (site density, land–sea mask) thecalculation of further idealized fingerprints is possible.

In the future, the extension of the method to threedimensions is planned. In this case the pronounced spa-tial anisotropy of the measurements has to be consid-ered, because the mean surface station distance isclearly larger than the vertical distance of measuringpoints of radiosondes. Further developments will leadto a four-dimensional version of the analysis schemetaking the temporal evolution of the fields into account.This means not only minimizing the spatial variabilityof the data but also the changes from one observationtime to the next during interpolation. A multivariateapproach seems to be a promising further development(e.g., for a better link between the mass and windfields). For such an expansion fingerprints seem topresent an ideal downscaling tool as they are multivari-ate. This fact allows modeling realistic temporal evolu-tions (e.g., the diurnal thermal oscillation).

Acknowledgments. Thanks to the Austrian ScienceFunds (FWF) for financial support under GrantsP12475 and P15079.

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A Mesoscale Data Analysis and Downscaling …...A Mesoscale Data Analysis and Downscaling Method over Complex Terrain REINHOLD STEINACKER,MATTHIAS RATHEISER,BENEDIKT BICA,BARBARA CHIMANI,MANFRED

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