††thanks: Equal Contribution††thanks: Equal Contribution

Current address:]Idaho State University, Pocatello, Idaho 83209 Current
address:]Universit‘a degli Studi di Brescia, 25123 Brescia, Italy

The CLAS Collaboration

# First Observation of Large Missing-Momentum $(e,e^{\prime}p)$ Cross-Section
Scaling and the onset of Correlated-Pair Dominance in Nuclei

I. Korover  Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  A.W. Denniston  Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  A. Kiral  Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  A. Schmidt  The George Washington
University, Washington DC 20052, USA  A. Lovato  Physics Division, Argonne
National Laboratory, Lemont, Illinois 60439, USA  N. Rocco  Theoretical
Physics Department, Fermi National Accelerator Laboratory
, P.O. Box 500, Batavia, Illinois 60510, USA  A. Nikolakopoulos  Theoretical
Physics Department, Fermi National Accelerator Laboratory
, P.O. Box 500, Batavia, Illinois 60510, USA  L.B. Weinstein  Old Dominion
University, Norfolk, Virginia 23529, USA  E. Piasetzky  School of Physics and
Astronomy, Tel Aviv University, Tel Aviv 69978, Israel  O. Hen<EMAIL_ADDRESS>Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
M.J. Amaryan  Old Dominion University, Norfolk, Virginia 23529, USA  Giovanni
Angelini  The George Washington University, Washington DC 20052, USA  H. Atac
Temple University, Philadelphia, PA 19122  N.A. Baltzell  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606, USA  L. Barion
INFN, Sezione di Ferrara, 44100 Ferrara, Italy  M. Battaglieri  Thomas
Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
INFN, Sezione di Genova, 16146 Genova, Italy  I. Bedlinskiy  National Research
Centre Kurchatov Institute - ITEP, Moscow, 117259, Russia  Fatiha Benmokhtar
Duquesne University, 600 Forbes Avenue, Pittsburgh, PA 15282  A. Bianconi
Universit‘a degli Studi di Brescia, 25123 Brescia, Italy  INFN, Sezione di
Pavia, 27100 Pavia, Italy  L. Biondo  INFN, Sezione di Genova, 16146 Genova,
Italy  INFN, Sezione di Catania, 95123 Catania, Italy  Universit‘a degli Studi
di Messina, 98166 Messina, Italy  A.S. Biselli  Fairfield University,
Fairfield CT 06824  Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
F. Bossù  IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France
S. Boiarinov  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  W.J. Briscoe  The George Washington University,
Washington DC 20052, USA  D. Bulumulla  Old Dominion University, Norfolk,
Virginia 23529, USA  V.D. Burkert  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606, USA  D.S. Carman  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606, USA  J.C.
Carvajal  Florida International University, Miami, Florida 33199  M. Caudron
Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  P.
Chatagnon  Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
T. Chetry  Mississippi State University, Mississippi State, MS 39762-5167  G.
Ciullo  INFN, Sezione di Ferrara, 44100 Ferrara, Italy  Universita’ di Ferrara
, 44121 Ferrara, Italy  L. Clark  University of Glasgow, Glasgow G12 8QQ,
United Kingdom  P.L. Cole  Lamar University, 4400 MLK Blvd, PO Box 10046,
Beaumont, Texas 77710  Catholic University of America, Washington, D.C. 20064
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606,
USA  M. Contalbrigo  INFN, Sezione di Ferrara, 44100 Ferrara, Italy  G.
Costantini  Universit‘a degli Studi di Brescia, 25123 Brescia, Italy  INFN,
Sezione di Pavia, 27100 Pavia, Italy  A. D’Angelo  INFN, Sezione di Roma Tor
Vergata, 00133 Rome, Italy  Universita’ di Roma Tor Vergata, 00133 Rome Italy
N. Dashyan  Yerevan Physics Institute, 375036 Yerevan, Armenia  R. De Vita
INFN, Sezione di Genova, 16146 Genova, Italy  M. Defurne  IRFU, CEA,
Université Paris-Saclay, F-91191 Gif-sur-Yvette, France  A. Deur  Thomas
Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA  S.
Diehl  II Physikalisches Institut der Universitaet Giessen, 35392 Giessen,
Germany  University of Connecticut, Storrs, Connecticut 06269  C. Djalali
Ohio University, Athens, Ohio 45701  M. Duer  Technische Universität
Darmstadt, Fachbereich Physik, Darmstadt, Germany  R. Dupre  Universit’e
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  H. Egiyan  Thomas
Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA  M.
Ehrhart  Physics Division, Argonne National Laboratory, Lemont, Illinois
60439, USA  A. El Alaoui  Universidad Técnica Federico Santa María, Casilla
110-V Valparaíso, Chile  L. El Fassi  Mississippi State University,
Mississippi State, MS 39762-5167  L. Elouadrhiri  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606, USA  P. Eugenio  Florida
State University, Tallahassee, Florida 32306  S. Fegan  University of York,
York YO10 5DD, United Kingdom  R. Fersch  Christopher Newport University,
Newport News, Virginia 23606  A. Filippi  INFN, Sezione di Torino, 10125
Torino, Italy  G. Gavalian  Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606, USA  University of New Hampshire, Durham, New
Hampshire 03824-3568  Y. Ghandilyan  Yerevan Physics Institute, 375036
Yerevan, Armenia  G.P. Gilfoyle  University of Richmond, Richmond, Virginia
23173  F.X. Girod  Thomas Jefferson National Accelerator Facility, Newport
News, Virginia 23606, USA  A.A. Golubenko  Skobeltsyn Institute of Nuclear
Physics, Lomonosov Moscow State University, 119234 Moscow, Russia  R.W. Gothe
University of South Carolina, Columbia, South Carolina 29208  K.A. Griffioen
College of William and Mary, Williamsburg, Virginia 23187-8795  M. Guidal
Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  L. Guo
Florida International University, Miami, Florida 33199  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606, USA  K. Hafidi
Physics Division, Argonne National Laboratory, Lemont, Illinois 60439, USA  H.
Hakobyan  Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso,
Chile  Yerevan Physics Institute, 375036 Yerevan, Armenia  M. Hattawy  Old
Dominion University, Norfolk, Virginia 23529, USA  T.B. Hayward  University of
Connecticut, Storrs, Connecticut 06269  D. Heddle  Christopher Newport
University, Newport News, Virginia 23606  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606, USA  K. Hicks  Ohio
University, Athens, Ohio 45701  A. Hobart  Universit’e Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France  M. Holtrop  University of New
Hampshire, Durham, New Hampshire 03824-3568  C.E. Hyde  Old Dominion
University, Norfolk, Virginia 23529, USA  Y. Ilieva  University of South
Carolina, Columbia, South Carolina 29208  The George Washington University,
Washington DC 20052, USA  D.G. Ireland  University of Glasgow, Glasgow G12
8QQ, United Kingdom  E.L. Isupov  Skobeltsyn Institute of Nuclear Physics,
Lomonosov Moscow State University, 119234 Moscow, Russia  H.S. Jo  Kyungpook
National University, Daegu 41566, Republic of Korea  K. Joo  University of
Connecticut, Storrs, Connecticut 06269  S. Joosten  Physics Division, Argonne
National Laboratory, Lemont, Illinois 60439, USA  D. Keller  University of
Virginia, Charlottesville, Virginia 22901  A. Khanal  Florida International
University, Miami, Florida 33199  M. Khandaker  [  Norfolk State University,
Norfolk, Virginia 23504  A. Kim  University of Connecticut, Storrs,
Connecticut 06269  W. Kim  Kyungpook National University, Daegu 41566,
Republic of Korea  A. Kripko  II Physikalisches Institut der Universitaet
Giessen, 35392 Giessen, Germany  V. Kubarovsky  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606, USA  Rensselaer
Polytechnic Institute, Troy, New York 12180-3590  L. Lanza  INFN, Sezione di
Roma Tor Vergata, 00133 Rome, Italy  M. Leali  Universit‘a degli Studi di
Brescia, 25123 Brescia, Italy  INFN, Sezione di Pavia, 27100 Pavia, Italy  P.
Lenisa  INFN, Sezione di Ferrara, 44100 Ferrara, Italy  Universita’ di Ferrara
, 44121 Ferrara, Italy  K. Livingston  University of Glasgow, Glasgow G12 8QQ,
United Kingdom  I .J .D. MacGregor  University of Glasgow, Glasgow G12 8QQ,
United Kingdom  D. Marchand  Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab,
91405 Orsay, France  L. Marsicano  INFN, Sezione di Genova, 16146 Genova,
Italy  V. Mascagna  [  Università degli Studi dell’Insubria, 22100 Como, Italy
INFN, Sezione di Pavia, 27100 Pavia, Italy  B. McKinnon  University of
Glasgow, Glasgow G12 8QQ, United Kingdom  S. Migliorati  Universit‘a degli
Studi di Brescia, 25123 Brescia, Italy  INFN, Sezione di Pavia, 27100 Pavia,
Italy  M. Mirazita  INFN, Laboratori Nazionali di Frascati, 00044 Frascati,
Italy  V. Mokeev  Thomas Jefferson National Accelerator Facility, Newport
News, Virginia 23606, USA  Skobeltsyn Institute of Nuclear Physics, Lomonosov
Moscow State University, 119234 Moscow, Russia  C. Munoz Camacho  Universit’e
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  P. Nadel-Turonski
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606,
USA  K. Neupane  University of South Carolina, Columbia, South Carolina 29208
S. Niccolai  Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
G. Niculescu  James Madison University, Harrisonburg, Virginia 22807  T. R.
O’Connell  University of Connecticut, Storrs, Connecticut 06269  M. Osipenko
INFN, Sezione di Genova, 16146 Genova, Italy  A.I. Ostrovidov  Florida State
University, Tallahassee, Florida 32306  P. Pandey  Old Dominion University,
Norfolk, Virginia 23529, USA  M. Paolone  New Mexico State University, PO Box
30001, Las Cruces, NM 88003, USA  L.L. Pappalardo  INFN, Sezione di Ferrara,
44100 Ferrara, Italy  Universita’ di Ferrara , 44121 Ferrara, Italy  R.
Paremuzyan  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  E. Pasyuk  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606, USA  O. Pogorelko  National Research
Centre Kurchatov Institute - ITEP, Moscow, 117259, Russia  M. Pokhrel  Old
Dominion University, Norfolk, Virginia 23529, USA  J. Poudel  Old Dominion
University, Norfolk, Virginia 23529, USA  J.W. Price  California State
University, Dominguez Hills, Carson, CA 90747  Y. Prok  Old Dominion
University, Norfolk, Virginia 23529, USA  University of Virginia,
Charlottesville, Virginia 22901  B.A. Raue  Florida International University,
Miami, Florida 33199  Thomas Jefferson National Accelerator Facility, Newport
News, Virginia 23606, USA  Trevor Reed  Florida International University,
Miami, Florida 33199  M. Ripani  INFN, Sezione di Genova, 16146 Genova, Italy
J. Ritman  Institute fur Kernphysik (Juelich), Juelich, Germany  A. Rizzo
INFN, Sezione di Roma Tor Vergata, 00133 Rome, Italy  Universita’ di Roma Tor
Vergata, 00133 Rome Italy  G. Rosner  University of Glasgow, Glasgow G12 8QQ,
United Kingdom  P. Rossi  Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606, USA  J. Rowley  Ohio University, Athens, Ohio
45701  F. Sabatié  IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette,
France  R.A. Schumacher  Carnegie Mellon University, Pittsburgh, Pennsylvania
15213  E.P. Segarra  Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Y.G. Sharabian  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606, USA  E.V. Shirokov
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
119234 Moscow, Russia  U. Shrestha  University of Connecticut, Storrs,
Connecticut 06269  O. Soto  INFN, Laboratori Nazionali di Frascati, 00044
Frascati, Italy  N. Sparveris  Temple University, Philadelphia, PA 19122  S.
Stepanyan  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  I.I. Strakovsky  The George Washington University,
Washington DC 20052, USA  S. Strauch  University of South Carolina, Columbia,
South Carolina 29208  The George Washington University, Washington DC 20052,
USA  R. Tyson  University of Glasgow, Glasgow G12 8QQ, United Kingdom  M.
Ungaro  Thomas Jefferson National Accelerator Facility, Newport News, Virginia
23606, USA  Rensselaer Polytechnic Institute, Troy, New York 12180-3590  L.
Venturelli  Universit‘a degli Studi di Brescia, 25123 Brescia, Italy  INFN,
Sezione di Pavia, 27100 Pavia, Italy  H. Voskanyan  Yerevan Physics Institute,
375036 Yerevan, Armenia  A. Vossen  Duke University, Durham, North Carolina
27708-0305  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  E. Voutier  Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab,
91405 Orsay, France  Kevin Wei  University of Connecticut, Storrs, Connecticut
06269  X. Wei  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  R. Wishart  University of Glasgow, Glasgow G12 8QQ,
United Kingdom  M.H. Wood  Canisius College, Buffalo, NY  University of South
Carolina, Columbia, South Carolina 29208  B. Yale  College of William and
Mary, Williamsburg, Virginia 23187-8795  N. Zachariou  University of York,
York YO10 5DD, United Kingdom  J. Zhang  University of Virginia,
Charlottesville, Virginia 22901  Z.W. Zhao  Duke University, Durham, North
Carolina 27708-0305

###### Abstract

We report the first measurement of $x_{B}$-scaling in $(e,e^{\prime}p)$ cross-
section ratios off nuclei relative to deuterium at large missing-momentum of
$350\leq p_{miss}\leq 600$ MeV/c. The observed scaling extends over a
kinematic range of $0.7\leq x_{B}\leq 1.8$, which is significantly wider than
$1.4\leq x_{B}\leq 1.8$ previously observed for inclusive $(e,e^{\prime})$
cross-section ratios. The $x_{B}$-integrated cross-section ratios become
constant (i.e., scale) beginning at $p_{miss}\approx k_{F}$, the nuclear Fermi
momentum. Comparing with theoretical calculations we find good agreement with
Generalized Contact Formalism calculations for high missing-momentum ($>375$
MeV/c), suggesting the observed scaling results from interacting with nucleons
in short-range correlated (SRC) pairs. For low missing-momenta, mean-field
calculations show good agreement with the data for $p_{miss}\leq k_{F}$, and
suggest that contributions to the measured cross-section ratios from
scattering off single, un-correlated, nucleons are non-negligible up to
$p_{miss}\approx 350$ MeV/c. Therefore, SRCs become dominant in nuclei at
$p_{miss}\approx 350$ MeV/c, well above the nuclear Fermi Surface of
$k_{F}\approx 250$ MeV/c.

Atomic nuclei are complex quantum-mechanical systems that account for most of
the visible mass in the universe. The complexity of the strong nuclear
interaction makes it difficult to use scattering reactions to experimentally
probe the detailed distributions of nucleons inside nuclei. Experimental
nuclear physicists thus work together with theorists to find measurable
reactions that are sensitive to particular aspects of nuclear dynamics.

By using high-energy electron beams to knock out nucleons from nuclei in
nearly elastic kinematics, one can learn about the behavior of single nucleons
in the nucleus Kelly (1996). This behavior can be generally explained by
nucleons moving in nuclear shell-model states (e.g. $s$-, $p$-, $d$-, …
shells) where the typical nucleon momenta in each shell is smaller than the
nuclear Fermi momentum ($k_{F}$). Full shell-model calculations improve on
this by introducing effective long-ranged correlations between the nucleons
Dickhoff and Barbieri (2004), which leads to the formation of a nuclear Fermi
Surface.

While these models can successfully describe the long-range structure of
nuclei, they do not describe the explicit high-resolution effects of short-
range correlated (SRC) nucleon pairs. Within a high-resolution picture of
nuclear dynamics, SRC pairs arise when two nucleons get so close to each other
that the short-range nuclear interaction between them is much stronger than
the effective long-ranged nuclear mean field due to their interactions with
all the other nucleons in the nucleus Frankfurt and Strikman (1988); Ciofi
degli Atti (2015).

SRCs have been clearly identified in data using large momentum-transfer
nucleon knockout reactions Tang et al. (2003); Subedi et al. (2008); Hen et
al. (2014a); Patsyuk et al. (2021); Hen et al. (2017). They are characterized
by a high (greater than $k_{F}$) relative momentum between the nucleons of the
pair and are predominantly proton-neutron pairs formed due to the action of
the spin-dependent tensor part of the strong nuclear interaction Schiavilla et
al. (2007); Sargsian et al. (2005); Alvioli et al. (2008); Neff et al. (2015).
They thus deplete the occupancy of single-particle shell-model states (below
$k_{F}$) and populate high-momentum states Frankfurt and Strikman (1988);
Egiyan et al. (2006); Ciofi degli Atti (2015); Hen et al. (2017); Paschalis et
al. (2020); Ryckebusch et al. (2019) While shell structures vary among nuclei,
SRCs are a universal phenomenon, i.e., they are similar in all nuclei Ciofi
degli Atti and Simula (1996); Ciofi degli Atti (2015); Ryckebusch et al.
(2015); Cruz-Torres et al. (2020), varying primarily in their magnitude.

A complete high-resolution microscopic description of atomic nuclei should
thus have the nucleus-dependent mean field and long-ranged nuclear shell model
parts as well as explicit nucleus-independent effects of SRC pairs.

Here we study the onset of SRC dominance in semi-inclusive high-energy
electron scattering reactions, where we detect the knocked-out proton in
addition to the scattered electron, $(e,e^{\prime}p)$. For the first time in
$(e,e^{\prime}p)$ reactions, we observed scaling in the cross-section ratios
of nuclei from carbon to lead relative to deuterium over a broad range in the
Bjorken scaling variable, $x_{B}$. This scaling substantially extends the
kinematical range where SRCs can be identified and studied, as compared with
previous inclusive $(e,e^{\prime})$ measurements. Thereby, they provide direct
experimental evidence for the dominance SRCs in the scattering response at
high missing momenta, and allow quantifying the onset of this dominance.

Our experiment ran at the Thomas Jefferson National Accelerator Facility. It
used a 5.01 GeV electron beam incident on a target system consisting of a
deuterium cell followed by an interchangeable solid foil of carbon (C),
aluminum (Al), iron (Fe), or lead (Pb) Hakobyan et al. (2008). Scattered
electrons and knocked-out protons were identified and measured using the CEBAF
Large Acceptance Spectrometer (CLAS) Mecking et al. (2003) (see supplementary
materials for details).

In high-energy scattering, the electron transfers a single virtual photon to
the nucleus with momentum $\vec{q}$ and energy $\omega$. In the high-
resolution quasielastic (QE) reaction picture, the virtual photon is absorbed
by a single nucleon, which gets knocked-out of the nucleus with momentum
$\vec{p}_{p}$. By measuring both the scattered electron and knocked-out
proton, i.e. the $(e,e^{\prime}p)$ reaction, we can determine the missing
momentum $\vec{p}_{miss}=\vec{p}_{p}-\vec{q}.$ The reaction is further
characterized by the four-momentum transfer
$Q^{2}=\vec{q}\thinspace^{2}-\omega^{2}$ and Bjorken scaling variable
$x_{B}=Q^{2}/2m\omega$ where $m$ is the nucleon mass.

If the knocked-out nucleon does not re-interact as it leaves the nucleus, then
$\vec{p}_{miss}$ is equal to the initial momentum of that nucleon. Thus we
expect the reaction to be sensitive to mean-field nucleons at low-${p}_{miss}$
and to SRCs at high-${p}_{miss}$ Duer et al. (2018). In the SRC dominated
region, the cross section ratio for any two nuclei should be constant (i.e.,
independent of $p_{miss}$) and equal to the relative number of high-momentum
nucleons in the two nuclei Frankfurt et al. (1993); Egiyan et al. (2003,
2006); Fomin et al. (2012); Schmookler et al. (2019); Ciofi degli Atti (2015);
Hen et al. (2017). Thus, by measuring the $(e,e^{\prime}p)$ cross section
ratio for nuclei relative to deuterium for different minimum ${p}_{miss}$, we
can establish the onset of scaling that corresponds to SRC pair dominance in
the nuclear momentum distribution.

Figure 1:  The intrinsic width of the 12C missing mass ($M_{miss}$)
distribution, plotted vs $x_{B}$. Black points show the data. The red curve
and uncertainty band shows an SRC based Generalized Contact Formalism (GCF)
calculationCruz-Torres et al. (2020); Pybus et al. (2020). The two main model
parameters of the calculation, namely SRC pair CM momentum distribution width
$\sigma_{CM}$ and the residual $A-2$ system excitation energy $E^{*}$, are fit
to the data. Data error bars and calculation error band show the total
uncertainty (statistical + systematical) at the $1\sigma$ or $68\%$ confidence
level. Inset: The resulting confidence intervals of the correlation between
the fitted values of $\sigma_{CM}$ and $E^{*}$. The inner region (red) shows
the $1\sigma$ ($68\%$) confidence region with each region increasing the
confidence by an additional $1\sigma$. The observed agreement between the data
and GCF calculation, and the agreement of the fitted model parameters with
previous extractions, show the measured $(e,e^{\prime}p)$ events are
consistent with resulting from the hard breakup of SRC pairs.

To study this, we measured the $(e,e^{\prime}p)$ reaction in conditions
sensitive to the knockout of protons from SRC pairs. We required $Q^{2}\geq
1.5$ (GeV/c)2 and $350\leq p_{miss}\leq 600$ MeV/c to ensure a high-resolution
reaction that can resolve single nucleons in SRC pairs. We further required
that the proton be emitted within $25^{\circ}$ of the momentum transfer, to
ensure that the measured proton was the nucleon that absorbed the virtual
photon Hen et al. (2014b); Schmidt et al. (2020).

We suppressed inelastic (non-QE) scattering events by requiring $M_{miss}$,
the missing mass for $(e,e^{\prime}p)$ scattering from a two-nucleon pair at
rest, to be smaller than the nucleon mass ($m$) plus pion mass ($m_{\pi}$),
$0.8\leq M_{miss}\leq m+m_{\pi}=1.08$ GeV/c2. In non-QE reactions the momentum
transferred to undetected particles (e.g., pions) shifts the direction of
$\vec{p}_{miss}$. Therefore such events will have a larger
$\theta_{\vec{p}_{miss},\vec{q}}$, the angle between $\vec{p}_{miss}$ and
$\vec{q}$. We thus further suppressed the small non-QE tail below
$M_{miss}=1.08$ GeV/c2 by observing that the measured
$\theta_{\vec{p}_{miss},\vec{q}}$ distribution has two maxima, corresponding
to QE and non-QE scattering, and selecting events in the
$\theta_{\vec{p}_{miss},\vec{q}}$ QE peak. See Figs. S1 and S2 and
supplementary materials for details.

We tested our identification of scattering from protons in SRC pairs by
comparing the measured width of the $M_{miss}$ peak with that calculated using
the Generalized Contact Formalism (GCF) model for SRC pairs (see Fig. 1) Weiss
et al. (2015, 2018); Cruz-Torres et al. (2020); Schmidt et al. (2020); Pybus
et al. (2020); Patsyuk et al. (2021). This width depends on the CLAS
resolution and on the SRC pair center of mass (CM) motion. We corrected for
the effects of the CLAS resolution by subtracting the deuterium $M_{miss}$
peak width from that of 12C to get the intrinsic width:
$\sigma^{{}^{12}C}_{int}=\sqrt{(\sigma^{{}^{12}C}_{exp})^{2}-(\sigma^{d}_{exp})^{2}}$.

The measured $x_{B}$ dependence of $\sigma^{{}^{12}C}_{int}$ agrees well with
a GCF calculation that assumes electron scattering from nucleons in SRC pairs
with a realistic Gaussian CM momentum distribution Cohen et al. (2018), as was
done in Refs. Schmidt et al. (2020); Pybus et al. (2020); Patsyuk et al.
(2021). The calculation accounts for the CLAS detector acceptance and
resolution and our event selection cuts. The width of the CM momentum
distributions, $\sigma_{CM}$, and the excitation energy of the residual
nuclear system after the SRC breakup, $E^{*}$, were the only two free
parameters used in the calculation and were determined from a fit of the
calculation to the data (see inset of Fig. 1). For $\sigma_{CM}$ the fitted
values of $160-210$ MeV/c ($125-220$) at $68\%$ ($90\%$) confidence agree well
with previous direct measurements Cohen et al. (2018); Patsyuk et al. (2021).
For $E^{*}$, while not previously measured, the fitted values of $20-55$ MeV
($0-70$ MeV) at $68\%$ ($90\%$) confidence are consistent with those used by
previous analyses Schmidt et al. (2020). The sensible values of the resulting
fit parameters and the agreement between the $x_{B}$ dependence of the GCF
calculation and the data further support our interpretation of the data as
dominated by scattering off SRC pairs.

Figure 2:  Measured $(e,e^{\prime}p)$ per nucleon cross-section ratios for
$350\leq p_{miss}\leq 600$ MeV/c for carbon, aluminum, iron and lead relative
to deuterium as function of $x_{B}$. Open squares are the inclusive
$(e,e^{\prime})$ per nucleon cross section ratios of Ref. Schmookler et al.
(2019). The horizontal lines show the average $(e,e^{\prime})$ cross section
ratio for $1.45\leq x_{B}\leq 1.8$ Schmookler et al. (2019). Error bars show
the data uncertainty (statistical plus point-to-point systematical) at the
$1\sigma$ or $68\%$ confidence level. Overall $(e,e^{\prime}p)$ systematic
uncertainties of 10% (C) to 15% (Pb) are not shown.

Using the selected event samples, we extracted the $(e,e^{\prime}p)$ cross
section ratios for scattering off the solid targets relative to deuterium. We
first divided the ratio of the measured numbers of events for a given target
to deuterium with the ratio of the experimentally determined integrated
luminosities to obtain the normalized-yield ratios. We then determined the
cross section ratios by correcting the normalized-yield ratios for attenuation
of the outgoing protons as they traverse the different nuclei Duer et al.
(2019a), electron radiation effects, and the small difference in the CLAS
acceptance for detecting particles emitted from the deuterium and the solid
targets. At the large $Q^{2}$ of this measurement, the attenuation correction
is less sensitive to the initial nucleon momenta and therefore both mean-field
and SRC breakup reactions have the same attenuation Duer et al. (2019a).
Acceptance effects were calculated using the CLAS detector simulation Wolin
(1996) and an electron scattering reaction event generator based on the GCF as
applied in previous studies Schmidt et al. (2020); Pybus et al. (2020) (see
supplementary materials for details).

Figure 2 shows the per nucleon $(e,e^{\prime}p)$ cross section ratios for
$350\leq{p}_{miss}\leq 600$ MeV/c for carbon, aluminum, iron, and lead
relative to deuterium as a function of $x_{B}$. The $(e,e^{\prime}p)$ ratios
scale (i.e., are constant) for all four nuclei over the entire measured
$x_{B}$ range. This implies that the reaction is probing similar nuclear
configurations in the measured nuclei and in deuterium. As the deuteron is a
simple proton-neutron correlated two-body system, we interpret this high
missing-momentum scaling as observation of deuteron-like proton-neutron SRC
pairs in nuclei. The cross-section ratio is thus a measure of their relative
abundance.

This interpretation is supported by the consistency between our measured
$(e,e^{\prime}p)$ cross section ratios and previously measured inclusive
$(e,e^{\prime})$ scattering cross section ratios at similar $Q^{2}$ and at
$x_{B}\geq 1.5$ Frankfurt et al. (1993); Egiyan et al. (2003, 2006); Fomin et
al. (2012); Schmookler et al. (2019) (see open symbols in Fig. 2). As the
inclusive scaling onset at $x_{B}\approx 1.5$ has been attributed to
scattering off nucleons with momenta greater than $\sim 275$ MeV/c Egiyan et
al. (2006); Weiss et al. (2021), it is also associated with scattering off
nucleons in deuteron-like proton-neutron SRC pairs, formed by the strong
tensor interaction Frankfurt et al. (1993); Schmookler et al. (2019) (see
supplementary materials for details). Proton detection extends the cross-
section ratio plateau down to $x_{B}=0.7$, providing a new experimental tool
to study the transition to SRC dominance in nuclei over a broad range in
$x_{B}$.

We next examined how this scaling depends on the minimum $p_{miss}$ range of
the data. Figure 3 shows the per nucleon $(e,e^{\prime}p)$ cross section
ratios for the measured nuclei relative to deuterium as a function of $x_{B}$
for different minimum $p_{miss}$. For all nuclei, the curve for
$p_{miss}^{min}=0$ are similar to the inclusive data of Schmookler et al.
Schmookler et al. (2019), with a plateau for $x_{B}\geq 1.5$ and a minimum at
$x_{B}\approx 1$. As $p_{miss}^{min}$ increases, this minimum fills in. For
$p_{miss}^{min}\gtrsim 200-250$ MeV/c, it is completely filled in and the
$(e,e^{\prime}p)$ cross section ratio scales over the full measured $x_{B}$
range of $0.7$ to $1.8$. This indicates that short-range interactions become
dominant at around $k_{F}\approx 220-260$ MeV/c Moniz et al. (1971), as
expected.

To better quantify this transition, Figure 4 shows the $p_{miss}$ dependence
of the $(e,e^{\prime}p)$ cross section ratio for the different nuclei relative
to deuterium, integrated over the scaling regions of $0.7\leq x_{B}\leq 1.8$.
The measured cross section ratio for carbon ($k_{F}\approx 220$ MeV/c),
aluminum ($k_{F}\approx 235$ MeV/c), and iron ($k_{F}\approx 260$ MeV/c) all
become flat starting around the Fermi momentum at $p_{miss}\approx 250$. The
lead ratio shows a similar transition but does not fully plateau, possibly
owing to its much larger neutron-to-proton ratio or to increased final state
interactions due to its larger size.

We thus conclude that the data indicate the existence of a clear transition in
the nuclear response around the nuclear Fermi momentum, resulted by the onset
of the SRC dominance at high-momenta.

Figure 3:  The per-nucleon cross-section ratios for carbon (left), aluminum,
iron and lead (right) to deuterium as a function of $x_{B}$. Full symbols with
different colors indicate different lower limits of the $p_{miss}$
integration. The upper $p_{miss}$ limit is fixed at $600$ MeV/c. The colored
bands mark the statistical plus point-to-point systematical uncertainty of the
data at the $\pm 1\sigma$ or $68\%$ confidence level. Overall systematic
uncertainties of 10–15% are not shown. Open black squares show the inclusive
$(e,e^{\prime})$ per nucleon cross-section ratios of Ref. Schmookler et al.
(2019). The horizontal line shows the average $(e,e^{\prime})$ cross section
ratio for $1.45\leq x_{B}\leq 1.8$ Schmookler et al. (2019).  Figure 4:  The
per nucleon $(e,e^{\prime}p)$ cross-section ratios for different nuclei to
deuterium as a function of $p_{miss}$, integrated over $0.7\leq x_{B}\leq
1.8$. Filled circles show the data. The teal, black, and azure lines show the
calculated cross sections obtained using QMC (teal), IPSM (black), and Skryme
(azure) based one-body mean-field models for nucleon distributions in carbon.
The brown line shows a GCF calculation for SRC nucleons in carbon. Data error
bars and the widths of the bands around the calculation lines show their
uncertainties (statistical plus point-to-point systematical) at the $1\sigma$
or $68\%$ confidence level. Overall data systematic data uncertainties of
10–15% are not shown. The insert shows the result of subtracting the mean-
field calculations from the carbon data.

Focusing on the carbon nucleus, where theoretical calculations are readily
available, we find that the high-$p_{miss}$ data are in excellent agreement
with an asymptotic GCF calculation of the cross-section ratio (brown band in
Fig. 4, left panel). The calculation was done using a factorized plane-wave
impulse approximation (PWIA) for the scattering reaction, with SRC-pair
spectral functions calculated using the GCF Weiss et al. (2019) and
transparency and single-charge exchange corrections as done in Ref. Duer et
al. (2019b); Schmidt et al. (2020); Pybus et al. (2020) (see supplementary
materials for details). The SRC-pair abundance parameters used by the GCF
calculation were all previously determined by $ab-initio$ many-body
theoretical calculations Cruz-Torres et al. (2020), offering additional
support to our identification of QE scattering events and the dominance of
interacting with nucleons in SRC pairs at high-$p_{miss}$.

Lastly, we estimate the possible contribution of single-nucleon (one-body)
states to the measured cross-section ratio around $k_{F}$, to assess their
impact on the observed SRC scaling onset. We examined three calculations using
different single-nucleon spectral functions: (1) Independent particle shell-
model with Woods-Saxon potential O’Neill et al. (1995); Makins et al. (1994),
(2) Skryme model using 5 different functionals Waroquier et al. (1979);
Chabanat et al. (1998) and (3) new Quantum Monte-Carlo (QMC) many-body
calculations of the overlap between the 12C and 11B+proton wave functions (see
supplementary materials for details). For the latter, we added the
contributions from both the ground state and a range of 11B excited states to
include a wide range of mean-field, single-nucleon states. These models each
assume very different underlying single-body nuclear dynamics, and thus the
spread in their results offers a general measure for the model dependence of
the single-body mean-field contribution.

The resulting calculated one-body mean-field contributions to the C/d
$(e,e^{\prime}p)$ cross section ratio are shown in Figure 4, left panel. The
calculations were done using the same factorized PWIA scheme as for the GCF
calculation, only using mean-field spectral functions. The IPSM and Skryme
calculations are re-normalized (quenched) to agree with our low-$p_{miss}$
($\leq 150$ MeV/c) high-$Q^{2}$ data. This effectively accounts for their lack
of single-nucleon strength lost to long- and short-ranged correlations and/or
few-body reaction operators that can compensate for it Tropiano et al. (2021).
In contrast, the QMC calculation extracts the underlying single-nucleon states
from the fully correlated high-resolution wave function. It thus has fewer
than six protons in its mean-field orbitals and does not require additional
quenching.. The agreement of the QMC calculation with the low-$p_{miss}$ data
thus confirms the completeness of the calculation.

The different single-nucleon calculations are similar and all show the
existence of residual single nucleon contributions above $k_{F}$. We
subtracted the calculated one-body mean-field contribution from the measured
cross-section ratio(see the inset in Fig. 4)left. Accounting for these
contributions can shift the scaling onset from the purely experimental onset
at $\sim k_{F}$ to a higher value of $\sim 350$ MeV/c. Such a shift would be
consistent with the existence of a nuclear Fermi-surface that accounts for
long-range correlations above $k_{F}$.

To conclude, the new nuclear scaling measurements presented herein allow
isolating interactions with SRC pairs in a substantially-extended kinematical
regime. By examining the scaling onset in missing momentum, we observe a
universal transition in the scattering response above the nuclear Fermi
momentum. Using model dependent estimates for mean field contributions, we see
an indication for the onset of full SRC dominance above $\sim 350$ MeV/c.
Detailed theoretical calculations will be able to use our data to fully
quantify this mean-field to SRC transition region and to obtain an effective
high-resolution description of a wide range of heavy nuclei.

###### Acknowledgements.

We acknowledge the efforts of the staff of the Accelerator and Physics
Divisions at Jefferson Lab that made this experiment possible. The analysis
presented here was carried out as part of the Jefferson Lab Hall B data-mining
project supported by the U.S. Department of Energy (DOE). The research was
also supported by the National Science Foundation, the Israel Science
Foundation, the Pazi Foundation, the Chilean Comisión Nacional de
Investigación Científica y Tecnológica, the French Centre National de la
Recherche Scientifique and Commissariat a l’Energie Atomique, the
French–American Cultural Exchange, the Italian Istituto Nazionale di Fisica
Nucleare, the National Research Foundation of Korea, and the UK Science and
Technology Facilities Council. Jefferson Science Associates operates the
Thomas Jefferson National Accelerator Facility for the DOE, Office of Science,
Office of Nuclear Physics under contract DE-AC05-06OR23177.

## References

  * Kelly (1996) J. Kelly, Adv. Nucl. Phys. 23, 75 (1996).
  * Dickhoff and Barbieri (2004) W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004), eprint nucl-th/0402034.
  * Frankfurt and Strikman (1988) L. Frankfurt and M. Strikman, Phys. Rep. 160, 235 (1988).
  * Ciofi degli Atti (2015) C. Ciofi degli Atti, Phys. Rept. 590, 1 (2015).
  * Tang et al. (2003) A. Tang et al., Phys. Rev. Lett. 90, 042301 (2003), eprint nucl-ex/0206003.
  * Subedi et al. (2008) R. Subedi et al., Science 320, 1476 (2008), eprint 0908.1514.
  * Hen et al. (2014a) O. Hen et al., Science 346, 614 (2014a), eprint 1412.0138.
  * Patsyuk et al. (2021) M. Patsyuk et al. (BM@N), Nature Physics 17, 693 (2021), eprint 2102.02626.
  * Hen et al. (2017) O. Hen, G. A. Miller, E. Piasetzky, and L. B. Weinstein, Rev. Mod. Phys. 89, 045002 (2017).
  * Schiavilla et al. (2007) R. Schiavilla, R. B. Wiringa, S. C. Pieper, and J. Carlson, Phys. Rev. Lett. 98, 132501 (2007).
  * Sargsian et al. (2005) M. M. Sargsian, T. V. Abrahamyan, M. I. Strikman, and L. L. Frankfurt, Phys. Rev. C 71, 044615 (2005).
  * Alvioli et al. (2008) M. Alvioli, C. Ciofi degli Atti, and H. Morita, Phys. Rev. Lett. 100, 162503 (2008).
  * Neff et al. (2015) T. Neff, H. Feldmeier, and W. Horiuchi, Phys. Rev. C 92, 024003 (2015).
  * Egiyan et al. (2006) K. Egiyan et al. (CLAS Collaboration), Phys. Rev. Lett. 96, 082501 (2006).
  * Paschalis et al. (2020) S. Paschalis, M. Petri, A. O. Macchiavelli, O. Hen, and E. Piasetzky, Phys. Lett. B 800, 135110 (2020), eprint 1812.08051.
  * Ryckebusch et al. (2019) J. Ryckebusch, W. Cosyn, T. Vieijra, and C. Casert, Phys. Rev. C 100, 054620 (2019), eprint 1907.07259.
  * Ciofi degli Atti and Simula (1996) C. Ciofi degli Atti and S. Simula, Phys. Rev. C 53, 1689 (1996).
  * Ryckebusch et al. (2015) J. Ryckebusch, M. Vanhalst, and W. Cosyn, Journal of Physics G: Nuclear and Particle Physics 42, 055104 (2015).
  * Cruz-Torres et al. (2020) R. Cruz-Torres et al., Nature Physics 17, 306 (2020), eprint 1907.03658.
  * Hakobyan et al. (2008) H. Hakobyan et al., Nucl. Instrum. Meth. A592, 218 (2008).
  * Mecking et al. (2003) B. A. Mecking et al., Nucl. Instrum. Meth. A503, 513 (2003).
  * Duer et al. (2018) M. Duer et al. (CLAS Collaboration), Nature 560, 617 (2018).
  * Frankfurt et al. (1993) L. Frankfurt, M. Strikman, D. Day, and M. Sargsyan, Phys. Rev. C 48, 2451 (1993).
  * Egiyan et al. (2003) K. Egiyan et al. (CLAS Collaboration), Phys. Rev. C 68, 014313 (2003).
  * Fomin et al. (2012) N. Fomin et al., Phys. Rev. Lett. 108, 092502 (2012).
  * Schmookler et al. (2019) B. Schmookler et al. (CLAS Collaboration), Nature 566, 354 (2019).
  * Pybus et al. (2020) J. Pybus, I. Korover, R. Weiss, A. Schmidt, N. Barnea, D. Higinbotham, E. Piasetzky, M. Strikman, L. Weinstein, and O. Hen, Phys. Lett. B 805, 135429 (2020), eprint 2003.02318.
  * Hen et al. (2014b) O. Hen et al., Science 346, 614 (2014b), eprint 1412.0138.
  * Schmidt et al. (2020) A. Schmidt et al. (CLAS), Nature 578, 540 (2020), eprint 2004.11221.
  * Weiss et al. (2015) R. Weiss, B. Bazak, and N. Barnea, Phys. Rev. C92, 054311 (2015), eprint 1503.07047.
  * Weiss et al. (2018) R. Weiss, R. Cruz-Torres, N. Barnea, E. Piasetzky, and O. Hen, Phys. Lett. B 780, 211 (2018).
  * Cohen et al. (2018) E. O. Cohen et al. (CLAS Collaboration), Phys. Rev. Lett. 121, 092501 (2018), eprint 1805.01981.
  * Duer et al. (2019a) M. Duer et al. (CLAS Collaboration), Phys. Lett. B797, 134792 (2019a), eprint 1811.01823.
  * Wolin (1996) E. Wolin, _Clas - geant simulation_ (1996), URL https://www.jlab.org/Hall-B/document/gsim/userguide.html.
  * Weiss et al. (2021) R. Weiss, A. W. Denniston, J. R. Pybus, O. Hen, E. Piasetzky, A. Schmidt, L. B. Weinstein, and N. Barnea, Phys. Rev. C 103, L031301 (2021), eprint 2005.01621.
  * Moniz et al. (1971) E. J. Moniz, I. Sick, R. R. Whitney, J. R. Ficenec, R. D. Kephart, and W. P. Trower, Phys. Rev. Lett. 26, 445 (1971).
  * Weiss et al. (2019) R. Weiss, I. Korover, E. Piasetzky, O. Hen, and N. Barnea, Phys. Lett. B791, 242 (2019), eprint 1806.10217.
  * Duer et al. (2019b) M. Duer et al. (CLAS Collaboration), Phys. Rev. Lett. 122, 172502 (2019b), eprint 1810.05343.
  * O’Neill et al. (1995) T. O’Neill, W. Lorenzon, P. Anthony, R. Arnold, J. Arrington, E. Beise, J. Belz, P. Bosted, H.-J. Bulten, M. Chapman, et al., Physics Letters B 351, 87 (1995), ISSN 0370-2693.
  * Makins et al. (1994) N. Makins et al., Phys. Rev. Lett. 72, 1986 (1994).
  * Waroquier et al. (1979) M. Waroquier, J. Sau, K. Heyde, P. Van Isacker, and H. Vincx, Phys. Rev. C 19, 1983 (1979).
  * Chabanat et al. (1998) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nucl. Phys. A 635, 231 (1998), [Erratum: Nucl.Phys.A 643, 441–441 (1998)].
  * Tropiano et al. (2021) A. J. Tropiano, S. K. Bogner, and R. J. Furnstahl, Phys. Rev. C 104, 034311 (2021), eprint 2105.13936.


# High-resolution monthly precipitation climatologies over Norway: assessment
of spatial interpolation methods

Alice Crespi  Department of Environmental Science and Policy, Università degli
Studi di Milano, Italy  Cristian Lussana  The Norwegian Meteorological
Institute, Oslo, Norway  Michele Brunetti  Istituto di Scienze dell’Atmosfera
e del Clima, CNR, Bologna, Italy  Andreas Dobler  The Norwegian Meteorological
Institute, Oslo, Norway  Maurizio Maugeri  Department of Environmental Science
and Policy, Università degli Studi di Milano, Italy  Istituto di Scienze
dell’Atmosfera e del Clima, CNR, Bologna, Italy  Ole Einar Tveito  The
Norwegian Meteorological Institute, Oslo, Norway

##  Abstract

Monthly precipitation climatologies at 1 km resolution have been produced over
the Norwegian mainland for 1981-2010. The observed station normals are
interpolated over a regular grid by applying a multi-linear local regression
kriging (MLRK). The statistical method aims at modeling the influence of the
main geographical features, such as: latitude, longitude, elevation and sea
nearness on the precipitation field at a local scale. The MLRK is composed of
two steps, (i) a background precipitation field is computed through a multi-
linear local regression based on the geographical information, then (ii) a
kriging interpolation is applied to adjust the field so to better fit the
station residuals (i.e., the difference between the observed normals and the
background field). The interpolation accuracy is evaluated by reconstructing
the station normals with a leave-one-out approach and by comparing the model
performances with those of other interpolation methods.
The poor observation coverage over remote and mountainous regions in Norway
has motivated us to consider precipitation fields produced by numerical
models. In fact, numerical model output provides a reference field for the
evaluation of MLRK that is not dependent on the station density, though it is
not as accurate as the observations. Specifically, a regional climate
simulation with a resolution of 2.5 km, based on the dynamical downscaling of
the global reanalysis ERAInterim and available for the time period 2003-2016,
has been used.
In addition to the MLRK validation, the combination of numerical model fields
and observed climatologies has been investigated and the results show that
this integrated approach provides more accurate high-resolution climatologies
over the Country. The numerical background introduces valuable information
over remote and mountainous regions, whereas in-situ data correct for model
biases.

##  1 Introduction

High-resolution gridded datasets of surface precipitation represent important
supporting tools for decision makers in a wide range of fields such as energy
production, management and conservation of natural sources as well as in
operative meteorology. In particular, due to their primary role in water
supply, hydrological balance and hydro-power production, gridded datasets of
monthly precipitation are essential products for Norway and they are required
in both climatological and hydrological applications.
In this document, a new version of 1981-2010 monthly precipitation
climatologies at 1 km resolution over Norway is presented. We aim at
representing small scale features in the climatologies, where the definition
of small scale depends on the local spatial distribution of observations. The
precipitation fields are based on in-situ observations only and statistical
methods have been applied for the spatial interpolation. The observations we
use have been quality checked and the raw data were considered, without
applying any correction for the wind-induced undercatch. The statistical
method we implemented takes into account the strong relationship between
rainfall and surface geographical features, that is well documented by the
scientific literature (e.g. Daly et al. (1994); Masson and Frei (2014)). More
precisely, monthly precipitation climatologies of Norway are interpolated by
the multi-linear local regression kriging (MLRK) in which gridded normals are
obtained by a two-step approach. First, a precipitation field is obtained by
means of a multi-linear local regression. For each grid point, the neighboring
stations are selected and a multi-linear function of their geographical
features (elevation, latitude, longitude and sea nearness) is fitted to the
correspondent precipitation normals. Second, the precipitation field obtained
in the first step is used as the background field for a residual kriging
procedure (Goovaerts (2000)), where the residuals between the observed normals
and the background values at station locations are spatially interpolated over
the high-resolution grid.
The observation network plays a crucial role in the determination of the
quality of the climatologies (Hofstra et al. (2008); Hofstra and New (2009)).
In dense station areas we get more accurate estimates while in data sparse
areas the final results are affected by a higher uncertainty. The Norwegian
station network is strongly uneven with higher station density in the South, a
sparser network in the North and a poor coverage over the mountainous areas,
especially at higher elevations. As a consequence, interpolation schemes, such
as those proposed in Daly et al. (2002) and Crespi et al. (2018) modeling
precipitation on a very local scale and using elevation as the main predictor
are more likely to produce unrealistic patterns over those complex and
scarcely observed areas. On the contrary, MLRK takes into account a larger
number of geographical predictors and it takes into account larger spatial
scales so that it is more robust with respect to inhomogeneities in the
spatial distribution of observations.
The monthly climatologies have been evaluated through cross-validation
methods. In particular, we have considered two domains, one in the north and
one in the south in order to study the influence of station density and
different climatic conditions on the model performances.
Our study focuses also on the comparison between observational gridded
datasets and numerical model output. Gridded numerical model data for hourly
precipitation is provided by the climate model version of HARMONIE (version
cy38h1.2), a seamless NWP model framework developed and used by several
national meteorological services. HARMONIE includes a set of different physics
packages adapted for different horizontal resolutions. For the high-
resolution, convection permitting simulations in this case, the model has been
set-up with AROME physics (Seity et al., 2011) and the SURFEX surface scheme
(Masson et al., 2013). The climate runs (hereafter referred to as "HCLIM-
AROME") have been carried out within the HARMONIE script system, covering the
period July 2003 to December 2016 on a 2.5 km grid over the Norwegian main
land. More details on the climate model can be found in Lind et al. (2016),
references therein and on _https://www.hirlam.org/trac/wiki/HarmonieClimate_.
Because the numerical model does not include measurements from the network of
rain-gauges, the output is an independent information that can be used to
evaluate observational datasets. Furthermore, the effective resolution of the
precipitation field obtained by numerical models can be considered rather
uniform over the spatial domain. For this reason, the numerical model output
can be used as a reference to assess the relative increase of uncertainty in
the observational gridded dataset while moving from data dense areas towards
data sparse areas.
The promising results obtained during our study, motivated us to investigate
the combination of HCLIM-AROME data and in-situ observations. In principle,
such a combination integrates the accuracy of the observations and the fine-
scale realistic patterns of the convection resolving numerical model. The MLRK
scheme has been modified so to include HCLIM-AROME monthly averaged
precipitation fields as the background for the residual kriging. This
combination turned out to improve the quality of average monthly precipitation
fields over the time period 2003-2016 that is covered by HCLIM-AROME, if
compared to the observational dataset. Therefore, the proposed combination
method can be used to obtain more accurate monthly precipitation normals over
Norway.

##  2 Observation data

The database considered for the construction of 1981-2010 monthly
precipitation climatologies for Norway includes more than 5000 daily
precipitation series covering Fennoscandia (Norway, Sweden and Finland) and
neighboring countries. Daily precipitation series were retrieved from ECA&D
archive https://www.ecad.eu//dailydata/index.php and the MET Norway Climate
database (KDVH). More precisely, ECA&D series relative to Norwegian sites were
replaced by MET stations if they were closer than 100 m (radial distance) and
1 m (altitude difference), while those MET series without a correspondence in
ECA&D archive were added to the database as new stations. When necessary,
ECA&D daily records were shifted in order to represent the accumulated
precipitation between 06 UTC of previous day to 06 UTC of current day, which
is the definition for daily precipitation in KDVH series. Moreover, metadata
containing geographical information (latitude, longitude and elevation) are
available for each series.
The database was subjected to a careful quality control procedure aiming at
detecting duplicate records, spurious data entries and correcting erroneous
station locations. In order to identify duplicates each series was compared to
the ones available in a radius of 15 km and common data in every year (only
values greater than 1 mm were considered) were checked for similarities. If
the fraction of identical entries was higher than 70% for a certain year, the
two series were put into correspondence. Single periods of duplicate values at
different site locations are likely to be due to erroneous data assignments
and they were invalidated, while in case of greater overlapping periods only
the longest and most continuous series was retained. In addition, in order to
avoid different data referring to the same geographical location, stations
showing equal coordinates were checked and their metadata corrected by means
of geographical information provided by the respective national meteorological
services or Google Earth.
After these procedures, monthly totals for each station were computed and
their data quality was evaluated on a monthly basis by comparing measured
records with a simulated series obtained by means on neighboring stations.
More precisely each monthly entry of each station (test) was simulated by
considering ten reference series. They were selected among the closest
stations which have enough common data (at least 9) for the month under
reconstruction with the test series and in which the entry to simulate is
available. The monthly value of test series ($p_{test_{m}})$ was computed from
the corresponding datum of each of the ten reference series ($p_{ref_{m},i})$
as follows:

$\tilde{p}_{test_{m},i}=p_{ref_{m},i}\cdot\frac{\bar{p}_{test_{m}}}{\bar{p}_{ref_{m},i}}\quad\quad(i=1,...,10)$
(1)

where $\bar{p}_{test_{m}}$ and $\bar{p}_{test_{m},i}$ are the test’s and
reference series’ averages for the considered month over their period of
common data. The final estimation of $p_{test_{m}}$ was finally obtained
considering the median of the ten values. High reconstruction errors, in terms
of mean absolute error (MAE) and root mean square error (RMSE), allowed to
point out and remove single problematic periods in a record or to discard
stations whose measured values completely mismatched the simulated ones.
After removing gross data errors, the daily series underwent a gap-filling
procedure aiming at maximizing the number and length of monthly data series
available for climatological purposes. The procedure was managed only if daily
gaps were contained in a month with valid entries for more than 2/3 of days
and for at least 50% of days in a 5-month period centered on it. For each gap
the stations in a radius of 50 km from the reconstructed one were considered
and correlation computed. A daily gap was reconstructed by considering the
value of the series showing the highest correlation (minimum threshold of 0.7)
and valid entries in common for at least 50% of the 5-month period. The daily
entry of reference series was reported to the mean daily value of the station
to fill by using as rescaling factor the ratio between their daily averages
over the common data period.
After fulfillment, monthly precipitation series were computed again for each
station. Whenever daily data were still missing, the corresponding monthly
total was not computed. On average, the filling procedure allowed to
reconstruct about 5% of missing daily data for each station, which
corresponded to a benefit in terms of monthly total precipitation computation
of about 15%. Monthly series relative to very close sites with contiguous and
homogeneous data were merged in order to maximize the data coverage over the
period chosen as reference for climatologies, while stations with less than 10
years of available monthly data, also after gap-filling and merging
activities, were definitely discarded from the database.
After these activities, 1981-2010 monthly precipitation normals were computed
for each series and, whenever this period was partially or completely
unavailable, missing months were reconstructed following the same method used
for quality-check (Eq. 1) but taking as final value the weighted mean of the
ten reference station estimates rather than their median. Since a remarkable
fraction of stations (52%) had more than 30% of missing data in the reference
period, this procedure allowed to prevent monthly normals being biased by a
lower fraction of monthly data entering in the average computation.
The resulting database available for the construction of Norwegian 1981-2010
high-resolution precipitation climatologies is composed by 3226 observation
sites covering Norway and surrounding countries as shown in figure 1. 1043
stations are located in Norway, even though they are unevenly distributed over
the domain: data coverage is generally higher in the South, especially around
the Oslo fjord, while observation availability decreases significantly in the
North especially over inland. Data availability gets even lower if high-
elevation areas are considered as pointed out in figures 1 and 2, where
distribution of available stations for elevation range is depicted. The
significant fraction of grid cells at high-elevation not covered by stations,
whose distribution strongly decrease especially above 1000 m a.s.l., suggests
that data coverage is not completely representative of study domain features.

Figure 1: Spatial distribution of available stations over Scandinavian
peninsula. Figure 2: Vertical distribution of the 1043 Norwegian stations
(solid line) compared to the grid cell elevation distribution (dashed line)
over Norway. The inset box shows the vertical distribution of the stations to
grid cells ratio.

##  3 Methods

###  3.1 Definition of interpolation grid

Precipitation climatologies for Norway are gridded on a smoothed version of 1
km x 1 km resolution Digital Elevation Model (DEM) expressed in Lambert
Azimuthal Equal-Area projection. This DEM version is retained by rescaling the
30 arc-second resolution GTOPO30 (USGS,1996) on a 1 km x 1 km grid and
assigning to each cell an elevation obtained as a weighted average of
elevations of surrounding cells. The station weight is expressed as a Gaussian
function decreasing to 0.5 at a distance of 3 km from the cell itself. The
smoothing degree is defined by varying the distance at which weights are
halved and choosing that minimizing the model errors (see below for details on
model errors). Smoothing of orographic details is performed in order to adapt
terrain representation to the actual scale at which precipitation-orography
interactions are supposed to occur (Foresti and Pozdnoukhov, 2011). Main
geographical parameters are computed for each cell ($x,y)$ of smoothed DEM
version:

  * •

slope orientation (facet)

$facet(x,y)=\begin{cases}\arccos\left(\frac{\frac{\partial{z}}{\partial{x}}}{|\vec{\nabla}z|}\right)&\text{if
}\left(\frac{\partial{z}}{\partial{x}}\right)>0\\\
2\pi-\arccos\left(\frac{\frac{\partial{z}}{\partial{x}}}{|\vec{\nabla}z|}\right)&\text{if
}\left(\frac{\partial{z}}{\partial{x}}\right)<0\end{cases}$ (2)

  * •

slope steepness

$Sl(x,y)=|\vec{\nabla}z|=\sqrt{\left(\frac{\partial{z}}{\partial{x}}\right)^{2}+\left(\frac{\partial{z}}{\partial{y}}\right)^{2}}$
(3)

  * •

distance from the sea is based on cross-distance computation which combines
horizontal distance and vertical gradients taking into account the presence of
orographic obstacles.

Geographical information for each station site, including smoothed elevation,
are finally extracted from the nearest DEM cell and used in interpolation
procedures described in next sections.

###  3.2 Interpolation model: multi-linear local regression kriging

High-resolution 1981-2010 monthly precipitation climatologies for Norway are
computed by performing Multi-linear Local Regression Kriging (MLRK) at each
cell of smoothed DEM. More precisely, this procedure is composed by two main
steps:

  1. 1.

multi-linear regression of precipitation versus chosen geographical predictors
is performed at each station site i and residual between observed and
estimated values is computed:

$\epsilon^{i}=p^{i}-\tilde{p}^{i}=p^{i}-\sum_{j=0}^{n}\alpha_{j}^{i}\cdot
q_{j}^{i}$ (4)

where n is the number of predictors, $\alpha$ is the vector of regression
coefficients and $q^{i}$ is the vector of predictors at target site. Latitude,
longitude, elevation and distance from the sea are chosen as regression
predictors while $\alpha$ are estimated for each month and for each station by
least-square method considering precipitation normals and geographical
features of all sample sites within 100 km from the station under
consideration. If less than 100 stations are available for regression within
this distance, searching radius is incremented of 10 km until the minimum
threshold is reached. Both the searching radius and the number of stations
entering in the regression were defined by the minimization of model errors
(see below for details on model errors).

  2. 2.

ordinary kriging (OK) of station residuals. In this case, experimental semi-
variogram is constructed by setting bin width to 15 km and maximum pair
distance to 300 km and by using exponential model as fitting curve.

Finally, precipitation at each grid cell is estimated:

$p(x,y)=\sum_{j=0}^{n}\alpha_{j}(x,y)\cdot
q_{j}(x,y)+\mathbf{k^{T}}(x,y)\cdot\epsilon$ (5)

where $\mathbf{k}(x,y)$ is the vector of kriging weights for cell $(x,y)$ and
$\alpha(x,y)$ are the regression coefficients estimated at grid point $(x,y)$
by considering its geographical features $q_{j}(x,y)$ and a minimum threshold
of 100 stations within 100 km from it.

###  3.3 The combination of HCLIM-AROME and observations

In order to evaluate the robustness of MLRK and its limits in handling with
areas with sparse station networks, statistical model outcomes were compared
with precipitation fields provided by a numerical model and a new
interpolation approach combining numerical model results with station
observations was tested. To this aim, monthly total precipitation series
produced by HCLIM-AROME on a 2.5 x 2.5 km grid covering Norway and a portion
of Sweden (figure 3) for 2003-2016 were considered.

Figure 3: HCLIM-AROME 2.5 km resolution field of mean Januar precipitation
2004-2016.

Monthly normals for each cell of 2.5 km grid were computed and the 12
resulting fields were then downscaled to the target 1 km resolution grid in
ETRS89-LAEA projection by means of bilinear interpolation routine provided by
the raster package in R software. In order to cover the same reference period,
monthly normals for 2003-2016 interval were computed for each station site and
used to provide MLRK gridded fields to be compared with HCLIM-AROME ones.
Moreover, high-resolution monthly precipitation climatologies were computed by
means of a kriging-based procedure combining HCLIM-AROME numerical model
predictions with station observations (HCLIM+KR). In HCLIM+KR, HCLIM-AROME
monthly field is substituted to the outcomes of multiple linear regression in
the first step of MLRK and it is used as background field in kriging
interpolation. More precisely, the residuals between station normals and the
values extracted from the closest HCLIM-AROME grid cell are modeled by OK and
results used to correct initial numerical field. Final precipitation
estimation at each grid cell $(x,y)$ is obtained as the sum of HCLIM-AROME
value at the point and corresponding kriging correcting coefficients:

$p(x,y)=p_{hclim}(x,y)+\mathbf{k_{hclim}^{T}}(x,y)\cdot\epsilon_{hclim}$ (6)

As in MLRK scheme, experimental variogram is constructed by setting bin width
and maximum pair distance to 15 km and 300 km, respectively, and theoretical
variogram is obtained by means of exponential fit.

##  4 Results

###  4.1 Model validation

MLRK performances were evaluated by comparing 1981-2010 monthly precipitation
normals of the 1043 Norwegian stations with the corresponding values estimated
by model in leave-one-out approach (LOO), i.e. by removing the value of
station under reconstruction in order to avoid self-influence. More precisely,
due to computational time reasons, the covariance matrix was computed from the
whole dataset, while the kriging weight of the station under consideration was
set to 0 and the remaining station weights were re-normalized.

MONTH | BIAS | MAE | RMSE | $R^{2}$
---|---|---|---|---
1 | 0.8 | 16.0 | 25.1 | 0.9
2 | 0.6 | 12.9 | 20.4 | 0.9
3 | 0.6 | 12.2 | 19.0 | 0.9
4 | 0.3 | 8.5 | 12.9 | 0.9
5 | 0.1 | 7.1 | 10.8 | 0.8
6 | 0.0 | 7.2 | 10.7 | 0.8
7 | 0.1 | 7.9 | 11.8 | 0.8
8 | 0.1 | 9.5 | 14.6 | 0.9
9 | 0.2 | 12.4 | 20.4 | 0.9
10 | 0.5 | 14.4 | 23.0 | 0.9
11 | 0.7 | 13.9 | 21.8 | 0.9
12 | 0.7 | 15.2 | 24.3 | 0.9
Table 1: Accuracy of MLRK monthly climatologies obtained from the leave-one-
out validation for the 1043 stations in Norway. All values are expressed in mm
and BIAS is defined as difference between simulation and observation.

In table 1 model errors obtained for all months are listed in terms of BIAS,
MAE, RMSE and $R^{2}$ as average over all validation stations. The highest
errors are obtained for winter months, especially in January, when the
greatest precipitation values occur over Norway, especially over the
mountainous region close to south-western coast. It is noteworthy that, as
depicted in figure 2, observation coverage decreases significantly on high-
elevation areas contributing to limit model ability in fully capture rainfall
gradients over orographically complex regions. However, $R^{2}$ is above 0.8
for all months, suggesting a generally good agreement between modeled and
measured values. Low values of monthly BIAS, even if systematically positive,
give evidence of no significant under or overestimation of station normals by
model reconstruction. In addition, the influence of station density on MLRK
accuracy was investigated by comparing LOO model errors obtained for Northern
sites (above Trondheim, 63∘18’ N) and Southern sites (below 63∘18’ N)
separately. The reconstruction errors, both in absolute and relative terms,
turn out to be 10% higher for the Northern stations than for the Southern ones
for all months. This discrepancy could be partly explained by the remarkable
difference in data coverage characterizing the two parts of the country.
In order to better verify the robustness of kriging applied to residuals of a
multiple regression approach and the suitability of chosen regression
predictors, we compared the results provided by other interpolation
approaches. More precisely, we took into account 1) an OK on the residuals of
a local regression of precipitation versus elevation and 2) a local weighted
linear precipitation-elevation regression (LWLR, Crespi et al. (2018)) in
which stations selected for the linear fit are weighted according to their
nearness and orographic similarities (elevation, sea distance, slope steepness
and slope orientation) with the point to estimate. In both tests, resulting
LOO errors are higher than those provided by MLRK for all months. The best
performance of MLRK could be firstly explained by the fact that the elevation
is not the leading forcing factor for precipitation distribution in Norway and
the integration of further geographical parameters into model equation
improves the representation of the spatial variability of precipitation over
the domain. Secondly, the uneven station distribution over Norway limits the
spatial scales that could be actually resolved by the models. This station
coverage is probably not suitable for interpolation approaches, such as LWLR,
focusing on the reconstruction of precipitation gradients on a too local scale
and it could lead to artifacts especially over remote areas characterizes by
rough terrains and low data availability. MLRK turns out to be more robust
over regions of sparse data coverage, since it performs the regression locally
but considering a larger spatial scale, while the resolvable small-scale
effect is then modeled by OK on regression residuals.

###  4.2 Comparison with 2003-2016 HCLIM-AROME dataset

Robustness and limits of MLRK in properly capturing the relationships between
rainfall and geographical features over Norway and in projecting them onto
high-resolution grid were further assessed by performing a comparison with
precipitation fields provided by HCLIM-AROME dataset. As already mentioned in
section 3.3, in order to perform comparison over the same time interval
spanned by HCLIM-AROME series, 2003-2016 station normals were computed.
Moreover, observation database was also redefined by including only the 2004
sites located inside area covered by HCLIM-AROME dataset. In order to evaluate
agreement between model results and station values, 2003-2016 monthly normals
of all 1043 Norwegian stations were compared to values of the closest cells in
the corresponding HCLIM-AROME grid. In table 2 errors are reported together
with those obtained by applying MLRK on the same station database of 2003-2016
monthly normals.

| HCLIM-AROME | MLRK
---|---|---
MONTH | BIAS | MAE | RMSE | BIAS | MAE | RMSE
1 | -21.5 | 26.5 | 38.1 | 0.4 | 15.0 | 23.1
2 | -10.3 | 17.2 | 24.5 | 0.2 | 11.9 | 18.0
3 | -12.6 | 20.0 | 28.3 | 0.3 | 12.1 | 18.8
4 | -4.0 | 15.4 | 20.9 | 0.2 | 9.3 | 14.1
5 | -2.5 | 14.6 | 19.5 | 0.0 | 8.4 | 12.2
6 | -2.3 | 17.2 | 22.3 | 0.0 | 7.8 | 11.0
7 | -1.0 | 20.4 | 27.2 | 0.1 | 8.3 | 11.7
8 | -8.4 | 22.8 | 30.3 | 0.0 | 9.9 | 14.3
9 | -21.2 | 28.5 | 41.0 | 0.0 | 13.7 | 22.3
10 | -24.2 | 28.8 | 38.9 | 0.2 | 13.6 | 21.6
11 | -23.6 | 31.5 | 44.3 | 0.4 | 15.7 | 24.7
12 | -23.2 | 30.3 | 43.7 | 0.4 | 17.7 | 28.2
Table 2: Accuracy of HCLIM-AROME monthly climatologies evaluated at station
locations and of MLRK reconstruction of 2003-2016 station normals in LOO
approach. In both cases validation subset is composed by 1043 Norwegian
stations, values are expressed in mm and BIAS is defined as difference between
model and observation.

HCLIM-AROME field shows greatest errors in all months and a systematically
negative bias, especially in winter season, corresponding to a general
underestimation of station normals. This tendency is better represented in
figure 4 showing the relative differences between HCLIM-AROME field and
observed normals at Norwegian station sites for January and July. HCLIM-AROME
monthly normals are lower than station values especially in coastal proximity,
while in inland rainfall values at station points are generally overestimated
by numerical model.

Figure 4: Relative differences between HCLIM-AROME field and station normals
for a) January and b) July.

Despite possible bias in precipitation estimates for certain areas,
pluviometric gradients depicted in HCLIM-AROME grids over regions without
stations are likely to be more robust than those provided by statistical model
since they are not affected by variation in data availability. Moreover, when
observations are used to improve numerical outcomes by means of kriging-based
distribution of station residuals described in section (HCLIM+KR), the
resulting LOO reconstruction errors decrease significantly in all months and
the bias is almost removed (table 3). This error improvement suggests that the
combination of resolved rainfall gradients of HCLIM-AROME fields with
observation corrections could compensate both the uneven coverage of the
station network and numerical model biases, thus providing more accurate
climatologies over Norway.

MONTH | BIAS | MAE | RMSE | $R^{2}$
---|---|---|---|---
1 | 0.3 | 12.9 | 19.4 | 0.9
2 | 0.2 | 10.4 | 15.7 | 0.9
3 | 0.2 | 10.5 | 16.3 | 0.9
4 | 0.1 | 8.1 | 12.3 | 0.9
5 | 0.1 | 7.7 | 10.9 | 0.9
6 | 0.0 | 7.6 | 10.4 | 0.8
7 | 0.1 | 8.7 | 12.2 | 0.8
8 | 0.1 | 9.5 | 13.2 | 0.9
9 | 0.2 | 11.6 | 18.5 | 0.9
10 | 0.2 | 11.8 | 18.3 | 0.9
11 | 0.3 | 13.7 | 21.8 | 0.9
12 | 0.3 | 16.0 | 25.0 | 0.9
Table 3: Accuracy of HCLIM+KR monthly climatologies obtained from the leave-
one-out validation for the 1043 stations in Norway. All values are expressed
in mm and BIAS is defined as difference between simulation and observation.

In addition to station point comparison, MLRK and HCLIM+KR differences were
evaluated also at grid cell level. In figures 5 and 6 the discrepancies
between precipitation distributions provided by the two approaches for January
and July are shown in both absolute and relative terms. While in July
differences mainly concerned very limited portions of inner areas located in
the highlands of Central and Southern Norway and range between -50% and +50%,
in January disagreement is more evident and distributed on larger portions all
over the domain. Besides systematically lower HCLIM+KR values along West-
Southern coast but with most deviations within 25%, the greatest differences
concern Southern inner highlands and mountain regions along Northern coasts,
where MLRK reconstructs lower precipitation values and deviations from
HCLIM+KR field overcome 80% for points at the highest elevations. In order to
further assess model differences, we clustered grid cells for classes of
elevation and we calculated the boxplot of model discrepancies for January
(figure 7). The boxplot points out the MLRK tendency to provide lower
precipitation normals than HCLIM+KR, especially at high-elevated grid cells.
This behavior could be due to the difficulties of MLRK in extrapolating the
actual precipitation-geographical relationship over complex regions where very
few stations are available and mainly located at low elevation and/or,
especially in Northern Norway, in coastal proximity. The role of data
distribution is even more clear if station points are superimposed on the map
of the model absolute differences for January (figure 8) proving that MLRK and
HCLIM+KR discrepancies increase with decreasing station density, especially
over reliefs. In addition, the variogram range (i.e. the maximum distance
within which stations are spatially correlated) resulting from the station
residuals of the multi-linear regression is generally below 20 km, while it is
greater than 80 km if station residuals from HCLIM-AROME field are considered.
The influence of observation residuals in correcting background field
decreases more rapidly with distance in MLRK approach and emphasizes its
difficulties to deal with remote regions.
The comparison, even if performed on a 14-year period of reference only,
allowed to assess that the statistical scheme based on MLRK provides
reasonable precipitation patterns over Norway. However, due to the uneven data
coverage of the domain MLRK accuracy could significantly decrease over areas
with very low station availability where actual pluviometric gradients could
not be fully captured by regression. The combination of precipitation fields
from a numerical model with precipitation normals derived from station
observations could represent a promising approach to enhance the
reconstruction of precipitation distribution also over Norwegian regions with
very poor data coverage.

Figure 5: Distribution of a) absolute and b) relative differences between MLRK
and HCLIM+KR fields for January. Figure 6: Distribution of a) absolute and b)
relative differences between MLRK and HCLIM+KR fields for July. Figure 7:
Absolute bias between MLRK and HCLIM+KR precipitation normals for January
clustered by elevation ranges. Figure 8: Distribution stations overlapped to
absolute differences between MLRK and HCLIM+KR fields for January.

##  5 1981-2010 monthly precipitation climatologies over Norway

In figures 9 and 10 MLRK seasonal and annual 1981-2010 Norwegian precipitation
climatologies at 1 km resolution are shown. A west-to-east gradient is evident
throughout the whole year with the highest precipitation values occurring
along the westernmost part of Norway and a clear transition to continental
regime towards inland regions. The coastal mountain ridge all along western
Norway acts as an important orographic barrier for moist air masses from the
sea which are quickly lifted up and loss their water vapor over a very short
distance from the coast. For this reason, the coastal regions experience the
greatest precipitation over the whole year, especially in south-western Norway
where the highest reliefs occurs and annual maxima of more than 3500 mm are
reconstructed. Areas featuring the lowest annual precipitation are located in
the northernmost mountainous part of the country and east of the mountain
ridge in the South with total amounts generally below 400 mm. On the seasonal
scale, wintertime is the wettest period with an average precipitation value of
302 mm and maxima of more than 1200 mm over the south-western mountain ridge,
especially in the area close to Sognefjorden. Spring turns to be, with about
190 mm as areal mean, the driest season when a large portion of inland
regions, especially the upper Gudbrandsdalen, features minimum precipitation
amounts and seasonal totals below 100 mm. It is noteworthy that, as explained
in previous sections, these areas, especially in the North, are also the least
covered by observations and gridded values are mostly extrapolated in MLRK by
using distant stations located in different climatic areas leading in some
cases to unreliable regression equations which emphasize dry conditions and
provide, even if for a very limited number of points, monthly values slightly
negative or very close to zero. Since these outcomes are intrinsic in the
modeling structure and could be removed only by increasing data coverage, grid
points with negative occurrences are retained and used to identify the areas
mostly affected by model uncertainty, while their negative estimations are set
to zero in the cumulate rainfall computation. Further information about
Norwegian climate could be retrieved by considering average yearly
precipitation cycles of main Norwegian subregions. As shown in figure 11, the
most remarkable distinction, also in terms of annual precipitation pattern,
regards coastal and inland regions. In particular, Norwegian areas which are
close to sea experience the highest relative monthly contributions to annual
precipitation in winter and autumn and the lowest ones in late spring and this
annual pattern turns out to be almost constant with latitude. On the contrary,
in inner regions, for which Finnmarkswidda and Østlandet are considered, the
main contributions to annual precipitation are provided by summer months while
minima are shifted to spring and early spring. Moreover, yearly rainfall cycle
of inlands shows a greater variation with latitude with an increase in summer
relative contributions of about 2% of the total moving Northern Norway. Oslo
Fjord represents an intermediate area with a smoother annual cycle showing the
lowest precipitation in early spring and the maximum contributions in late
summer when convective phenomena are dominant.

Figure 9: Seasonal distribution of 1981-2010 precipitation normals over Norway
gridded onto 1 km resolution by MLRK. Figure 10: Annual distribution of
1981-2010 precipitation normals over Norway gridded onto 1 km resolution by
MLRK. Figure 11: Average yearly precipitation cycles of main Norwegian
subregions. Values on y-axis are expressed as relative monthly contributions
to total annual precipitation.

##  6 Conclusions

A new version of 1981-2010 monthly precipitation climatologies at 1 km
resolution has been produced over Norway. The climatologies are based on ECA&D
and KDVH daily observation databases over the Scandinavian peninsula. As a
preliminary step, the station data have been quality-checked and a daily-gap
filling procedure has been applied. The interpolation of 1981-2010 station
monthly normals has been performed by means of a multi-linear local regression
kriging (MLRK). The statistical interpolation method combines a local
regression of station precipitation normals versus several geographical
predictors with a kriging interpolation of the station regression residuals.
The geographical parameters considered are elevation, latitude, longitude and
distance from the sea. The MLRK performances have been evaluated by studying
the summary statistics derived from a leave-one-out cross-validation approach
of the station monthly normals. All the Norwegian stations have been
considered as the basis for the cross-validation. The monthly errors, in terms
of MAE, range between 7.1 mm in May and 16.0 mm in January, while $R^{2}$ is
above 0.8 in all months. MLRK have been compared to other statistical
interpolation approaches, such as a local weighted linear precipitation-
elevation regression (LWLR) and a regression kriging considering only
elevation as precipitation predictor. MLRK turns out to be preferable than the
other interpolation approaches over Norway. In fact, the spatial scale
considered by MLRK fits better with the uneven station distribution of the
domain and the inclusion of more than one regression predictor allows to
improve the model estimations.
MLRK gridded precipitation have been also compared to HCLIM-AROME fields
derived from a 2003-2016 reanalysis driven regional climate model with a
resolution of 2.5 km. The numerical model provides realistic precipitation
fields, that are independent from the in-situ observations. On the other hand,
HCLIM-AROME estimates show higher biases than MLRK, especially over coastal
locations. A new combined interpolation approach (HCLIM+KR) has been
implemented and evaluated over the 2003-2016 period. In this procedure the
HCLIM-AROME fields are used as the background for the kriging interpolation of
the station residuals so that the gridded precipitation is estimated by means
of both observations and numerical output. If compared to the MLRK cross-
validation results, this integrated product leads to a significant decrease of
validation errors and it represents a valid approach to increase the accuracy
of the gridded precipitation climatologies over Norway, provided the
availability of HCLIM-AROME fields covering longer periods than 2003-2016.
According to the 1981-2010 monthly precipitation climatologies computed by
MLRK over Norway, the annual and seasonal maps show a strong west-to-south
gradient occurring between the wet mountainous coastal regions and the
continental climate of the outback. The annual totals range between about 4000
mm over the west coast and less than 300 mm in the southern inner areas and in
Finnmarkswidda. The comparison of the mean annual cycles of six Norwegian
subdomains proves the opposite climate regime occurring between coasts and
inlands but it allows also to depict more in detail the specific subregional
features of annual precipitation patterns over the country.

## References

  * Crespi et al. (2018) Crespi, A., M. Brunetti, G. Lentini, and M. Maugeri (2018), 1961–1990 high-resolution monthly precipitation climatologies for italy, International Journal of Climatology, 38(2), 878–895.
  * Daly et al. (1994) Daly, C., R. P. Neilson, and D. L. Phillips (1994), A statistical-topographic model for mapping climatological precipitation over mountainous terrain, Journal of Applied Meteorology, 33(2), 140–158.
  * Daly et al. (2002) Daly, C., W. P. Gibson, G. H. Taylor, G. L. Johnson, and P. Pasteris (2002), A knowledge-based approach to the statistical mapping of climate, Climate Research, 22(2), 99–113.
  * Foresti and Pozdnoukhov (2011) Foresti, L., and A. Pozdnoukhov (2011), Exploration of alpine orographic precipitation patterns with radar image processing and clustering techniques, Meteorological Applications, 19(4), 407–419.
  * Goovaerts (2000) Goovaerts, P. (2000), Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall, Journal of Hydrology, 228(1), 113 – 129.
  * Hofstra and New (2009) Hofstra, N., and M. New (2009), Spatial variability in correlation decay distance and influence on angular-distance weighting interpolation of daily precipitation over europe, International Journal of Climatology, 29(12), 1872–1880.
  * Hofstra et al. (2008) Hofstra, N., M. Haylock, M. New, P. Jones, and C. Frei (2008), Comparison of six methods for the interpolation of daily, european climate data, Journal of Geophysical Research: Atmospheres, 113(D21).
  * Lind et al. (2016) Lind, P., D. Lindstedt, E. Kjellström, and C. Jones (2016), Spatial and temporal characteristics of summer precipitation over central europe in a suite of high-resolution climate models, Journal of Climate, 29(10), 3501–3518, doi:10.1175/JCLI-D-15-0463.1.
  * Masson and Frei (2014) Masson, D., and C. Frei (2014), Spatial analysis of precipitation in a high-mountain region: exploring methods with multi-scale topographic predictors and circulation types, Hydrology and Earth System Sciences, 18(11), 4543–4563.
  * Masson et al. (2013) Masson, V., P. Le Moigne, E. Martin, S. Faroux, A. Alias, R. Alkama, S. Belamari, A. Barbu, A. Boone, F. Bouyssel, P. Brousseau, E. Brun, J.-C. Calvet, D. Carrer, B. Decharme, C. Delire, S. Donier, K. Essaouini, A.-L. Gibelin, H. Giordani, F. Habets, M. Jidane, G. Kerdraon, E. Kourzeneva, M. Lafaysse, S. Lafont, C. Lebeaupin Brossier, A. Lemonsu, J.-F. Mahfouf, P. Marguinaud, M. Mokhtari, S. Morin, G. Pigeon, R. Salgado, Y. Seity, F. Taillefer, G. Tanguy, P. Tulet, B. Vincendon, V. Vionnet, and A. Voldoire (2013), The surfexv7.2 land and ocean surface platform for coupled or offline simulation of earth surface variables and fluxes, Geoscientific Model Development, 6(4), 929–960, doi:10.5194/gmd-6-929-2013.
  * Seity et al. (2011) Seity, Y., P. Brousseau, S. Malardel, G. Hello, P. Bénard, F. Bouttier, C. Lac, and V. Masson (2011), The arome-france convective-scale operational model, Monthly Weather Review, 139(3), 976–991, doi:10.1175/2010MWR3425.1.