  We perform a detailed analysis on broad pulses in bright Gamma-ray bursts
(GRBs) to understand the evolution of GRB broad pulses. Using the temporal and
spectral properties, we test the high latitude emission (HLE) scenario in the
decaying phase of broad pulses. The HLE originates from the curvature effect of
a relativistic spherical jet, where higher latitude photons are delayed and
softer than the observer's line-of-sight emission. The signature of HLE has not
yet been identified undisputedly during the prompt emission of GRBs. The HLE
theory predicts a specific relation, F$_{\nu, E_{p}}$ $\propto$ E$_{p}\!^{2}$,
between the peak energy $E_{p}$ in $\nu$F$_{\nu}$ spectra and the spectral flux
F$_{\nu}$ measured at $E_{p}$, F$_{\nu, E_{p}}$. We search for evidence of this
relation in 2157 GRBs detected by the Gamma-ray Burst Monitor (GBM) on board
the Fermi Gamma-ray Space Telescope (Fermi) from the years 2008 to 2017. After
imposing unbiased selection criteria in order to minimize contamination in a
signal by background and overlaps of pulses, we build a sample of 32 broad
pulses in 32 GRBs. We perform a time-resolved spectral analysis on each of
these 32 broad pulses and find that the evolution of 18 pulses (56%) is clearly
consistent with the HLE relation. For the 18 broad pulses, the exponent
$\delta$ in the relation of F$_{\nu, E_{p}}$ $\propto$ E$_{p}\!^{\delta}$ is
distributed as a Gaussian function with median and width of 1.99 and 0.34,
respectively. This result provides constraint on the emission radius of GRBs
with the HLE signature.
